武汉大学数学与统计学院：线性代数习题全解：线性代数习题全集

分类：数学格式：pdf 日期：2008年12月18日

L?}
(]6
1?)5? Is?
?
Email,huangzh@whu.edu.cn
óqvD
DDd9Dy,μóq430072
Wuhan University
"
c
?Bc?

T1
?=c ?#
17
??c ?￥?MDDL?Z?F33
?
1c_
F￥L?M1?48
??cM
?#=Q?69
?Bc?

T
[a￥5′¤á
ì-%ù.'cyp×? A[/5,5.(2),(5); 7; 8.(2),(?+?5|y ?μ?) y¤.)?
]?Cμz￥3E,h?
(á?.
1, ?¨?LE59
/
?¨?

T:
(1)
flfl
flfl
flfl
flfl
2 0 1
1?4?1
1 8 3
flfl
flfl
flfl
flfl; (2)
flfl
flfl
flfl
flfl
a b c
b c a
c a b
flfl
flfl
flfl
flfl;
(3)
flfl
flfl
flfl
flfl
1 1 1
a b c
a2 b2 c2
flfl
flfl
flfl
flfl; (4)
flfl
flfl
flfl
flfl
x y x+y
y x+y x
x+y x y
flfl
flfl
flfl
flfl
.
3,(1)
flfl
flfl
flfl
flfl
2 0 1
1?4?1
1 8 3
flfl
flfl
flfl
flfl
= 2￡(?4)￡3+0￡(?1)￡(?1)+1￡1￡8?0￡1￡3?2￡(?1)￡8?1￡(?4)￡(?1)
=?24+8+16?4
=?4:
(2) flflfl
flfl
flfl
fl
a b c
b c a
c a b
flfl
flfl
flfl
flfl
= acb+bac+cba?bbb?aaa?ccc = 3abc?a3?b3?c3:
(3) flflfl
flfl
flfl
fl
1 1 1
a b c
a2 b2 c2
flfl
flfl
flfl
flfl
= bc2 +ca2 +ab2?ac2?ba2?cb2 = (a?b)(b?c)(c?a):
(4) fl
flfl
flfl
flfl
fl
x y x+y
y x+y x
x+y x y
flfl
flfl
flfl
flfl
= x(x+y)y +yx(x+y)+(x+y)yx?y3?(x+y)3?x3
= 3xy(x+y)?y3?3x2y?3y2x?x3?y3?x3
=?2(x3 +y3):
2,?1?
Vl?v1SQ?,p/
ò
￥
I?
:
(1) 1 2 3 4; (2) 4 1 3 2;
(3) 3 4 2 1; (4) 2 4 1 3;
(5) 1 3￠￠￠(2n?1) 2 4￠￠￠(2n);
(6) 1 3￠￠￠(2n?1) (2n) (2n?2)￠￠￠2.
3
(1)
I?
10.
(2)
I?
14,4 1,4 3,4 2,3 2.
1
2?Bc?

T
(3)
I?
15,3 2,3 1,4 2,4 1,2 1.
(4)
I?
13,2 1,4 1,4 3.
(5)
I?
1n(n?1)2,
3 2...........................................................................1?
5 2,5 4,...................................................................,2?
7 2,7 4,7 6..................................................................3?
..................................................................................
(2n?1) 2,(2n?1) 4,(2n?1) 6,...,(2n?1) (2n?2)..............(n?1)?
(6)
I?
1n(n?1):
3 2...........................................................................1?
5 2,5 4,...................................................................,2?
7 2,7 4,7 6..................................................................3?
..................................................................................
(2n?1) 2,(2n?1) 4,(2n?1) 6,...,(2n?1) (2n?2)..............(n?1)?
4 2...........................................................................1?
6 2,6 4,....................................................................,2?
..................................................................................
(2n) 2,(2n) 4,(2n) 6,...,(2n) (2n?2),..............................(n?1)?
3,
1¨?

T?cμy0a11a23￥[.
3:??l?,
1¨?

T￥B?[1
(?1)ta1p1a2p2a3p3a4p4;
?t1p1p2p3p4￥
I?

??p1 = 1,p2 = 3X%?,p1p2p3p4o
??13,'13241342.?￥
I?
tsY1
0+0+1+0 = 1;0+0+0+2 = 2:
[,?a11a23a32a44a11a23a34a421
p.
4,9
/
ò?

T:
(1)
flfl
flfl
flfl
flfl
flfl
4 1 2 4
1 2 0 2
10 5 2 0
0 1 1 7
flfl
flfl
flfl
flfl
flfl; (2)
flfl
flfl
flfl
flfl
flfl
2 1 4 1
3?1 2 1
1 2 3 2
5 0 6 2
flfl
flfl
flfl
flfl
flfl;
(3)
flfl
flfl
flfl
flfl
ab ac ae
bd?cd de
bf cf?ef
flfl
flfl
flfl
flfl; (4)
flfl
flfl
flfl
flfl
flfl
a 1 0 0
1 b 1 0
0?1 c 1
0 0?1 d
flfl
flfl
flfl
flfl
flfl
.
3,(1) flfl
flfl
flfl
flfl
flfl
4 1 2 4
1 2 0 2
10 5 2 0
0 1 1 7
flfl
flfl
flfl
flfl
flfl
r1$r2======?
flfl
flfl
flfl
flfl
flfl
1 2 0 2
4 1 2 4
10 5 2 0
0 1 1 7
flfl
flfl
flfl
flfl
flfl
r2?4r1=======
r3?10r1?
flfl
flfl
flfl
flfl
flfl
1 2 0 2
0?7 2?4
0?15 2?20
0 1 1 7
flfl
flfl
flfl
flfl
flfl
L?}
(]6
1?)5? Is?3
r2$r4======
flfl
flfl
flfl
flfl
flfl
1 2 0 2
0 1 1 7
0?15 2?20
0?7 2?4
flfl
flfl
flfl
flfl
flfl
r4+7r2=======
r3+15r2
flfl
flfl
flfl
flfl
flfl
1 2 0 2
0 1 1 7
0 0 17 85
0 0 9 45
flfl
flfl
flfl
flfl
flfl
= 17￡9
flfl
flfl
flfl
flfl
flfl
1 2 0 2
0 1 1 7
0 0 1 5
0 0 1 5
flfl
flfl
flfl
flfl
flfl
= 0:
(2) flfl
flfl
flfl
flfl
flfl
2 1 4 1
3?1 2 1
1 2 3 2
5 0 6 2
flfl
flfl
flfl
flfl
flfl
c4?c2=====
flfl
flfl
flfl
flfl
flfl
2 1 4 0
3?1 2 2
1 2 3 0
5 0 6 2
flfl
flfl
flfl
flfl
flfl
r4?r2=====
flfl
flfl
flfl
flfl
flfl
2 1 4 0
3?1 2 2
1 2 3 0
2 1 4 0
flfl
flfl
flfl
flfl
flfl
r4?r1=====
flfl
flfl
flfl
flfl
flfl
2 1 4 0
3?1 2 2
1 2 3 0
0 0 0 0
flfl
flfl
flfl
flfl
flfl
= 0:
(3) flflfl
flfl
flfl
fl
ab ac ae
bd?cd de
bf cf?ef
flfl
flfl
flfl
flfl
= adf
flfl
flfl
flfl
flfl
b c e
b?c e
b c?e
flfl
flfl
flfl
flfl
= adfbce
flfl
flfl
flfl
flfl
1 1 1
1?1 1
1 1?1
flfl
flfl
flfl
flfl
r2+r1=====
r3+r1 adfbce
flfl
flfl
flfl
flfl
1 1 1
0 0 2
0 2 0
flfl
flfl
flfl
flfl
=?adfbce
flfl
flfl
fl
0 2
2 0
flfl
flfl
fl = 4abcdef:
(4) flfl
flfl
flfl
flfl
flfl
a 1 0 0
1 b 1 0
0?1 c 1
0 0?1 d
flfl
flfl
flfl
flfl
flfl
r1+ar2======
flfl
flfl
flfl
flfl
flfl
0 1+ab a 0
1 b 1 0
0?1 c 1
0 0?1 d
flfl
flfl
flfl
flfl
flfl
??1
========
Z 7
(?1)(?1)2+1
flfl
flfl
flfl
flfl
1+ab a 0
1 c 1
0?1 d
flfl
flfl
flfl
flfl
c3+dc2======
flfl
flfl
flfl
flfl
1+ab a ad
1 c 1+cd
0?1 0
flfl
flfl
flfl
flfl
??3?========
Z 7
(?1)(?1)3+2
flfl
flfl
fl
1+ab ad
1 1+cd
flfl
flfl
fl = abcd+ab+cd+ad+1:
5,￡
ü:
(1)
flfl
flfl
flfl
flfl
a2 ab b2
2a a+b 2b
1 1 1
flfl
flfl
flfl
flfl
= (a?b)3;
￡
üflflfl
flfl
flfl
fl
a2 ab b2
2a a+b 2b
1 1 1
flfl
flfl
flfl
flfl
c2?c1=====
c3?c1
flfl
flfl
flfl
flfl
a2 ab?a2 b2?a2
2a b?a 2b?2a
1 0 0
flfl
flfl
flfl
flfl
=(?1)3+1
flfl
flfl
fl
ab?a2 b2?a2
b?a 2b?2a
flfl
flfl
fl = (b?a)(b?a)
flfl
flfl
fl
a b+a
1 2
flfl
flfl
fl = (a?b)
3:
(2)
flfl
flfl
flfl
flfl
ax+by ay +bz az +bx
ay +bz az +bx ax+by
az +bx ax+by ay +bz
flfl
flfl
flfl
flfl
= (a3 +b3)
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl;
4?Bc?

T
￡
ü:flfl
flfl
flfl
flfl
ax+by ay +bz az +bx
ay +bz az +bx ax+by
az +bx ax+by ay +bz
flfl
flfl
flfl
flfl
??1
========
s
7
a
flfl
flfl
flfl
flfl
x ay +bz az +bx
y az +bx ax+by
z ax+by ay +bz
flfl
flfl
flfl
flfl
+b
flfl
flfl
flfl
flfl
y ay +bz az +bx
z az +bx ax+by
x ax+by ay +bz
flfl
flfl
flfl
flfl
Q====
7
a2
flfl
flfl
flfl
flfl
x ay +bz z
y az +bx x
z ax+by y
flfl
flfl
flfl
flfl
+0+0+b2
flfl
flfl
flfl
flfl
y z az +bx
z x ax+by
x y ay +bz
flfl
flfl
flfl
flfl
Q====
7
a3
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
+b3
flfl
flfl
flfl
flfl
y z x
z x y
x y z
flfl
flfl
flfl
flfl
=a3
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
+b3(?1)2
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
= (a3 +b3)
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
:
N5μB?\ü?"￥3E:
flfl
flfl
flfl
flfl
ax+by ay +bz az +bx
ay +bz az +bx ax+by
az +bx ax+by ay +bz
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
ax ay az
ay az ax
az ax ay
flfl
flfl
flfl
flfl
+
flfl
flfl
flfl
flfl
by bz bx
bz bx by
bx by bz
flfl
flfl
flfl
flfl
=a3
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
+b3
flfl
flfl
flfl
flfl
y z x
z x y
x y z
flfl
flfl
flfl
flfl
= a3
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
+b3(?1)2
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
=(a3 +b3)
flfl
flfl
flfl
flfl
x y z
y z x
z x y
flfl
flfl
flfl
flfl
:
??3E\ A
?
",
L5
^B?p3!á
ì<?,?

T?
T??.?.
T.
￥FE:
flfl
flfl
flfl
flfl
a11,:,a1n
...,..,..
an1 ￠￠￠ ann
flfl
flfl
flfl
flfl
+
flfl
flfl
flfl
flfl
b11,:,b1n
...,..,..
bn1 ￠￠￠ bnn
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
a11 +b11,:,a1n +b1n
...,..,..
an1 +bn1 ￠￠￠ ann +bnn
flfl
flfl
flfl
flfl
:
(3)
flfl
flfl
flfl
flfl
flfl
a2 (a+1)2 (a+2)2 (a+3)2
b2 (b+1)2 (b+2)2 (b+3)2
c2 (c+1)2 (c+2)2 (c+3)2
d2 (d+1)2 (d+2)2 (d+3)2
flfl
flfl
flfl
flfl
flfl
= 0;
￡
ü,flfl
flfl
flfl
flfl
flfl
a2 (a+1)2 (a+2)2 (a+3)2
b2 (b+1)2 (b+2)2 (b+3)2
c2 (c+1)2 (c+2)2 (c+3)2
d2 (d+1)2 (d+2)2 (d+3)2
flfl
flfl
flfl
flfl
flfl
cj?c1======
j=2;3;4
flfl
flfl
flfl
flfl
flfl
a2 2a+1 4a+4 6a+9
b2 2b+1 4b+4 6b+9
c2 2c+1 4c+4 6c+9
d2 2d+1 4d+4 6d+9
flfl
flfl
flfl
flfl
flfl
c3?2c2======
c4?3c2
flfl
flfl
flfl
flfl
flfl
a2 2a+1 2 6
b2 2b+1 2 6
c2 2c+1 2 6
d2 2d+1 2 6
flfl
flfl
flfl
flfl
flfl

?1 è========= 0:
L?}
(]6
1?)5? Is?5
(4)
flfl
flfl
flfl
flfl
flfl
1 1 1 1
a b c d
a2 b2 c2 d2
a4 b4 c4 d4
flfl
flfl
flfl
flfl
flfl
= (a?b)(a?c)(a?d)(b?c)(b?d)(c?d)(a+b+c+d);
￡
ü:
flfl
flfl
flfl
flfl
flfl
1 1 1 1
a b c d
a2 b2 c2 d2
a4 b4 c4 d4
flfl
flfl
flfl
flfl
flfl
cj?c1======
j=2;3;4
flfl
flfl
flfl
flfl
flfl
1 0 0 0
a b?a c?a d?a
a2 b2?a2 c2?a2 d2?a2
a4 b4?a4 c4?a4 d4?a4
flfl
flfl
flfl
flfl
flfl
Z 7r1======
flfl
flfl
flfl
flfl
b?a c?a d?a
b2?a2 c2?a2 d2?a2
b2(b2?a2) c2(c2?a2) d2(d2?a2)
flfl
flfl
flfl
flfl
= (b?a)(c?a)(d?a)
flfl
flfl
flfl
flfl
1 1 1
b+a c+a d+a
b2(b+a) c2(c+a) d2(d+a)
flfl
flfl
flfl
flfl
c2?c1=====
c3?c1 (b?a)(c?a)(d?a)
flfl
flfl
flfl
flfl
1 0 0
b+a c?b d?b
b2(b+a) c2(c+a)?b2(b+a) d2(d+a)?b2(b+a)
flfl
flfl
flfl
flfl
Z 7r1====== (b?a)(c?a)(d?a)(c?b)(d?b)
flfl
flfl
fl
1 1
(c2 +bc+b2)+a(c+b) (d2 +bd+b2)+a(d+b)
flfl
flfl
fl
= (a?b)(a?c)(a?d)(b?c)(b?d)(c?d)(a+b+c+d):
(5)
flfl
flfl
flfl
flfl
flfl
flfl
fl
x?1 0 ￠￠￠ 0 0
0 x?1 ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
an an?1 an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
fl
= xn +a1xn?1 +￠￠￠+an?1x+an.
￡
ü:ZEB.
!Eü??L
￥xM10,??B
Z 7.
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
x?1 0 ￠￠￠ 0 0 0
0 x?1 ￠￠￠ 0 0 0
...,..,..,..,..,..
0 0 0 ￠￠￠ x?1 0
0 0 0 ￠￠￠ 0 x?1
an an?1 an?2 ￠￠￠ a3 a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
cn?1+xcn========
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
x?1 0 ￠￠￠ 0 0 0
0 x?1 ￠￠￠ 0 0 0
...,..,..,..,..,..
0 0 0 ￠￠￠ x?1 0
0 0 0 ￠￠￠ 0 0?1
an an?1 an?2 ￠￠￠ a3 x2 +a1x+a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
6?Bc?

T
cn?2+xcn?1==========
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
x?1 0 ￠￠￠ 0 0 0
0 x?1 ￠￠￠ 0 0 0
...,..,..,..,..,..
0 0 0 ￠￠￠ 0?1 0
0 0 0 ￠￠￠ 0 0?1
an an?1 an?2 ￠￠￠ x3 +a1x3 +a2x+a3 x2 +a1x+a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
cj+xcj?1========
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
0?1 ￠￠￠ 0 0
0 0 ￠￠￠ 0 0
...,..,..,..
0 0 ￠￠￠?1 0
0 0 ￠￠￠ 0?1
xn +a1xn?1 +￠￠￠+an?1x+an xn?1 +a1xn?2 +￠￠￠+an?2x+an?1 ￠￠￠ x2 +a1x+a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
=(xn +a1xn?1 +￠￠￠+an?1x+an)(?1)n+1
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 ￠￠￠ 0 0
0 ￠￠￠ 0 0
...,..,..
0 ￠￠￠?1 0
0 ￠￠￠ 0?1
flfl
flfl
flfl
flfl
flfl
flfl
fl
=(xn +a1xn?1 +￠￠￠+an?1x+an)(?1)n+1(?1)n?1
=xn +a1xn?1 +￠￠￠+an?1x+an:
ZE=.
!Eü?1??M10,¤?B?/?? ?.
?x = 0,5Dn = an.?
T? ?.
?x 6= 0,5
Dn c2+
1
xc1=======
flfl
flfl
flfl
flfl
flfl
flfl
fl
x 0 0 ￠￠￠ 0 0
0 x?1 ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
an an?1 + anx an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
fl
c3+1xc2=======
flfl
flfl
flfl
flfl
flfl
flfl
fl
x 0 0 ￠￠￠ 0 0
0 x 0 ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
an an?1 + anx an?2 + an?1x + anx2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
fl
=￠￠￠
=
flfl
flfl
flfl
flfl
flfl
flfl
fl
x 0 0 ￠￠￠ 0 0
0 x 0 ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x 0
an an?1 + anx an?2 + an?1x + anx2 ￠￠￠ P2 P1
flfl
flfl
flfl
flfl
flfl
flfl
fl
? ú,
P2 = a2 + a3x + a4x2 +￠￠￠++ anxn?2;
L?}
(]6
1?)5? Is?7
P1 = x+a1 + a2x + a3x2 +￠￠￠++ anxn?1:
¤?/???,
[
Dn = xn?1 ￠P1 = xn +a1xn?1 +￠￠￠+an?1x+an:
ZE?.¨?BE￡
ü.:
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
x?1 0 ￠￠￠ 0 0
0 x?1 ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
an an?1 an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
fl;
5
DnZ 7c1======x
flfl
flfl
flfl
flfl
flfl
fl
x?1 ￠￠￠ 0 0
...,..,..,..
0 0 ￠￠￠ x?1
an?1 an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
fl
+an(?1)n+1
flfl
flfl
flfl
flfl
flfl
fl
1 0 ￠￠￠ 0 0
x?1 ￠￠￠ 0 0
...,..,..,..
0 0 ￠￠￠ x?1
flfl
flfl
flfl
flfl
flfl
fl
= xDn?1 +an(?1)n+1(?1)n?1 = xDn?1 +an:
[,Dn = xDn?1 +an.?N?B
T¤
Dn = xn +a1xn?1 +￠￠￠+an?1x+an:
ZE
1.?KaB?Z 7.5 Aan?i￥}
?0
T.y1
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
x?1
x?1
x,..
..,?1
x?1
x?1
x,..
..,?1
x?1
an an?1 an?2 ￠￠￠ an?(i?1) an?i an?(i+1) a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
?an?i
￥?
￥
,P
?
^i￡i￥Z v,·/?
^(n?i?1)￡(n?i?1)￥Z v,?/?10.
5an?i￥::}::
::?::0::
T1(?i?an?i)?n?ai+1
)
(?1)n+i+1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
x?1
x?1
x,..
..,?1
x
flfl
flfl
flfl
flfl
flfl
flfl
flfl
i￡i
flfl
flfl
flfl
flfl
flfl
fl
1
x,..
..,?1
x?1
flfl
flfl
flfl
flfl
flfl
fl
(n?i?1)￡(n?i?1)
= xi
[,Dn?KaB?Z 7,¤?
Dn = an +an?1x+an?2x2 +￠￠￠+an?ixi +￠￠￠+a2xn?2 +(x+a1)xn?1
8?Bc?

T
= xn +a1xn?1 +￠￠￠+an?1x+an:
ZE?.?c1TMD.
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
x?1 0 ￠￠￠ 0 0
0 x?1 ￠￠￠ 0 0
0 0 x ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
an an?1 an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
c1+xc2======
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
0?1 0 ￠￠￠ 0 0
x2 x?1 ￠￠￠ 0 0
0 0 x ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
an +an?1x an?1 an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
c1+x2c3=======
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
0?1 0 ￠￠￠ 0 0
0 x?1 ￠￠￠ 0 0
x3 0 x ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
an +an?1x+an?2x2 an?1 an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
=￠￠￠
=
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
0?1 0 ￠￠￠ 0 0
0 x?1 ￠￠￠ 0 0
0 0 x ￠￠￠ 0 0
...,..,..,..,..
0 0 0 ￠￠￠ x?1
P an?1 an?2 ￠￠￠ a2 x+a1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl;
? ú,P = an +an?1x+an?2x2 +￠￠￠+a1xn?1 +xn.
??B
Z 7,¤
Dn = xn +a1xn?1 +￠￠￠+an?1x+an:
6,
!n¨?

TD = det(aij),üD
/Ja
I
H?è90–aG??LJ,GQ¤
D1 =
flfl
flfl
flfl
flfl
an1 ￠￠￠ ann
...,..
a11 ￠￠￠ a1n
flfl
flfl
flfl
flfl; D2 =
flfl
flfl
flfl
flfl
a1n ￠￠￠ ann
...,..
a11 ￠￠￠ an1
flfl
flfl
flfl
flfl; D3 =
flfl
flfl
flfl
flfl
ann ￠￠￠ a1n
...,..
an1 ￠￠￠ a11
flfl
flfl
flfl
flfl;
￡
üD1 = D2 = (?1)n(n?1)2 D,D3 = D.
￡
ü:
D1 =
flfl
flfl
flfl
flfl
an1 ￠￠￠ ann
...,..
a11 ￠￠￠ a1n
flfl
flfl
flfl
flfl
n?1Q?￥M
#oD===============
PrnD??B?
(?1)n?1
flfl
flfl
flfl
flfl
flfl
fl
a11 ￠￠￠ a1n
an1 ￠￠￠ ann
...,..
a21 ￠￠￠ a2n
flfl
flfl
flfl
flfl
flfl
fl
L?}
(]6
1?)5? Is?9
n?2Q?￥M
#oD===============
PrnD??=?
(?1)n?1(?1)n?2
flfl
flfl
flfl
flfl
flfl
flfl
fl
a11 ￠￠￠ a1n
a21 ￠￠￠ a2n
an1 ￠￠￠ ann
...,..
a31 ￠￠￠ a3n
flfl
flfl
flfl
flfl
flfl
flfl
fl
= ￠￠￠
= (?1)n?1(?1)n?2￠￠￠(?1)
flfl
flfl
flfl
flfl
a11 ￠￠￠ a1n
...,..
an1 ￠￠￠ ann
flfl
flfl
flfl
flfl
= (?1)1+2+￠￠￠+(n?2)+(n?1)D = (?1)n(n?1)2 D:
] ? V￡
D2 = (?1)n(n?1)2
flfl
flfl
flfl
flfl
a11 ￠￠￠ an1
...,..
a1n ￠￠￠ ann
flfl
flfl
flfl
flfl
= (?1)n(n?1)2 DT = (?1)n(n?1)2 D:
D3 = (?1)n(n?1)2 D2 = (?1)n(n?1)2 (?1)n(n?1)2 D = (?1)n(n?1)D = D:
7,9
/
ò?

T(Dk1k¨?

T)
(1) Dn =
flfl
flfl
flfl
flfl
a 1
...
1 a
flfl
flfl
flfl
flfl
,??L
í
í?
^a,?￥í
í?
^0;
3:ZEB.|cnTn?1Q
￥M
#D,M??=
:
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
a 0 ￠￠￠ 0 1
0 a ￠￠￠ 0 0
...,..,..,..
0 0 ￠￠￠ a 0
1 0 ￠￠￠ 0 a
flfl
flfl
flfl
flfl
flfl
flfl
fl
= (?1)n?1
flfl
flfl
flfl
flfl
flfl
flfl
fl
a 0 ￠￠￠ 0 1
1 0 ￠￠￠ 0 a
0 a ￠￠￠ 0 0
...,..,..,..
0 0 ￠￠￠ a 0
flfl
flfl
flfl
flfl
flfl
flfl
fl
|rnTn?1Q?￥M
#D,M??=?:
Dn = (?1)n?1(?1)n?1
flfl
flfl
flfl
flfl
flfl
flfl
fl
a 1 0 ￠￠￠ 0
1 a 0 ￠￠￠ 0
0 0 a ￠￠￠ 0
...,..,..,..
0 0 0 ￠￠￠ a
flfl
flfl
flfl
flfl
flfl
flfl
fl
=
flfl
flfl
fl
a 1
1 a
flfl
flfl
fl
flfl
flfl
flfl
flfl
a ￠￠￠ 0
...,..
0 ￠￠￠ a
flfl
flfl
flfl
flfl
(n?2)￡(n?2)
= (a2?1)an?2:
ZE=.
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
a 0 ￠￠￠ 0 1
0 a ￠￠￠ 0 0
...,..,..,..
0 0 ￠￠￠ a 0
1 0 ￠￠￠ 0 a
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c1====== a
flfl
flfl
flfl
flfl
a
...
a
flfl
flfl
flfl
flfl
(n?1)￡(n?1)
+1￡(?1)n+1
flfl
flfl
flfl
flfl
flfl
fl
0 ￠￠￠ 0 1
a ￠￠￠ 0 0
...,..,..
0 ￠￠￠ a 0
flfl
flfl
flfl
flfl
flfl
fl
(n?1)￡(n?1)
10?Bc?

T
Z 7r1====== an +(?1)n+1 ￡1￡(?1)(n?1)+1
flfl
flfl
flfl
flfl
a
...
a
flfl
flfl
flfl
flfl
(n?2)￡(n?2)
= an?an?2:
(2) Dn =
flfl
flfl
flfl
flfl
flfl
fl
x a ￠￠￠ a
a x ￠￠￠ a
...,..,..
a a ￠￠￠ x
flfl
flfl
flfl
flfl
flfl
fl;
3:ZEB.|?B?e(?1)sYF??ò?,¤
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
x a a ￠￠￠ a
a?x x?a 0 ￠￠￠ 0
a?x 0 x?a ￠￠￠ 0
...,..,..,..
a?x 0 0 ￠￠￠ x?a
flfl
flfl
flfl
flfl
flfl
flfl
fl;
|ò
?F??B

,¤
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
x+(n?1)a a a ￠￠￠ a
0 x?a 0 ￠￠￠ 0
0 0 x?a ￠￠￠ 0
...,..,..,..
0 0 0 ￠￠￠ x?a
flfl
flfl
flfl
flfl
flfl
flfl
fl
= ￡x+(n?1)a?(x?a)n?1:
ZE=.|ò
?F??B
¤
Dn =
flfl
flfl
flfl
flfl
flfl
fl
x+(n?1)a a ￠￠￠ a
x+(n?1)a x ￠￠￠ a
...,..,..
x+(n?1)a a ￠￠￠ x
flfl
flfl
flfl
flfl
flfl
fl
= ￡x+(n?1)a?
flfl
flfl
flfl
flfl
flfl
fl
1 a ￠￠￠ a
1 x ￠￠￠ a
...,..,..
1 a ￠￠￠ x
flfl
flfl
flfl
flfl
flfl
fl
|?B?e[(?1)sYF??ò?,¤
Dn = ￡x+(n?1)a?
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 a a ￠￠￠ a
0 x?a 0 ￠￠￠ 0
0 0 x?a ￠￠￠ 0
...,..,..,..
0 0 0 ￠￠￠ x?a
flfl
flfl
flfl
flfl
flfl
flfl
fl
= ￡x+(n?1)a?(x?a)n?1:
ZE?.
6¨E.
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 a a ￠￠￠ a
0 x a ￠￠￠ a
0 a x ￠￠￠ a
...,..,..,..
0 a a ￠￠￠ x
flfl
flfl
flfl
flfl
flfl
flfl
fl
(n+1)￡(n+1)
ri?r1=======
i=2;3;￠￠￠
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 a a ￠￠￠ a
1 x?a 0 ￠￠￠ 0
1 0 x?a ￠￠￠ 0
...,..,..,..
1 0 0 ￠￠￠ x?a
flfl
flfl
flfl
flfl
flfl
flfl
fl
(n+1)￡(n+1)
L?}
(]6
1?)5? Is?11
?x = a,5Dn = 0.?x 6= a,5|1x?acjF?c1,j = 2;3;￠￠￠ ;n+1:
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
1+ ax?an a a ￠￠￠ a
0 x?a 0 ￠￠￠ 0
0 0 x?a ￠￠￠ 0
...,..,..,..
0 0 0 ￠￠￠ x?a
flfl
flfl
flfl
flfl
flfl
flfl
fl
(n+1)￡(n+1)
=
1+ nax?a
(x?a)n = ￡x+(n?1)a?(x?a)n?1:
(3) Dn+1 =
flfl
flfl
flfl
flfl
flfl
flfl
fl
an (a?1)n ￠￠￠ (a?n)n
an?1 (a?1)n?1 ￠￠￠ (a?n)n?1
...,..,..
a a?1 ￠￠￠ a?n
1 1 ￠￠￠ 1
flfl
flfl
flfl
flfl
flfl
flfl
fl; (4
U: ?¨S￡
￡?

T￥2T,)
3:V?n+1? 7
S,?n+1?üVnQM
#D,D??1?;?n?ü(n?1)QDD??2
?.ün+(n?1)+￠￠￠+1 = n(n+1)2Q??D,¤(?°¤?56￥2
)
Dn+1 = (?1)n(n+1)2
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 1 ￠￠￠ 1
a a?1 ￠￠￠ a?n
...,..,..
an?1 (a?1)n?1 ￠￠￠ (a?n)n?1
an (a?1)n ￠￠￠ (a?n)n
flfl
flfl
flfl
flfl
flfl
flfl
fl;
N?

T1S￡
￡?

T.
vS￡
￡?

T￥E?,? ú￥a = x1,a? 1 = x2,...,a? (n? 1) = xn,a? n = xn+1.5
xi = a?(i?1),xj = a?(j?1).
[
Dn+1 = (?1)n(n+1)2
Y
n+1>i>j>1
x
i?xj
￠
= (?1)n(n+1)2
Y
n+1>i>j>1
￡(a?i+1)?(a?j +1)?
= (?1)n(n+1)2
Y
n+1>i>j>1
￡?(i?j)?
= (?1)n(n+1)2 ￡(?1)n+(n?1)+￠￠￠+1 ￡
Y
n+1>i>j>1
(i?j)
=
Y
n+1>i>j>1
(i?j):
(4) D2n =
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
fl
an 0 bn
...,..
0 a1 b1
c1 d1
0
...,..
cn 0 dn
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
fl;
3:ZEB.|c2nT2n?1Q
￥M
#D,M??=
;|r2nT2n?1Q?￥M
#D,M?
12?Bc?

T
?=?:
D2n = (?1)2n?1(?1)2n?1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
an bn
cn dn
an?1 bn?1
...,..
a1 b1
c1 d1
...,..
cn?1 dn?1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
= (andn?bncn)D2(n?1);
?n = 1
HD2 =
flfl
flfl
fl
a1 b1
c1 d1
flfl
flfl
fl = a1d1?b1c1,
[
D2n = (andn?bncn)￠￠￠(a1d1?b1c1) =
nY
i=1
(aidi?bici):
??ZED??P.15￥ è11M].'5￥?(1)l59¨?
NZE.
ZE=.
D2nZ 7r1====== an
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
fl
an?1 0 bn?1 0
...,..
0 a1 b1
c1 d1
0,..
...,..
cn?1 0 dn?1 0
0 ￠￠￠ 0 dn
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
fl
+(?1)2n+1bn
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
fl
0 an?1 0 bn?1
...,..
0,.,a1 b1c
1 d1
0
...,..
0 cn?1 dn?1
cn 0 0 0
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c2n?1========= a
ndnD2n?2?bncnD2n?2:
?N¤?w
T:
D2n = (andn?bncn)D2n?2;
'D2n =
nY
i=2
(aidi?bici)D2.
7D2 =
flfl
flfl
fl
a1 b1
c1 d1
flfl
flfl
fl = a1d1?b1c1,¤D2n =
nY
i=1
(aidi?bici).
(5) Dn = det(aij),?aij = ji?jj;
3:?aij = ji?jj¤
Dn = det(aij) =
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
0 1 2 3 ￠￠￠ n?1
1 0 1 2 ￠￠￠ n?2
2 1 0 1 ￠￠￠ n?3
...,..,..,..,..
n?2 n?3 n?4 n?5 ￠￠￠ 1
n?1 n?2 n?3 n?4 ￠￠￠ 0
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
L?}
(]6
1?)5? Is?13
ri?ri+1=======
i=1;2;￠￠￠
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
1 1 1 1 ￠￠￠ 1
1?1 1 1 ￠￠￠ 1
1?1?1 1 ￠￠￠ 1
...,..,..,..,..
1?1?1?1 ￠￠￠ 1
n?1 n?2 n?3 n?4 ￠￠￠ 0
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
cj+c1=======
j=2;3;￠￠￠
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
1 0 0 0 ￠￠￠ 0
1?2 0 0 ￠￠￠ 0
1?2?2 0 ￠￠￠ 0
...,..,..,..,..
1?2?2?2 ￠￠￠ 0
n?1 2n?3 2n?4 2n?5 ￠￠￠ n?1
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
=(?1)n?1(n?1)2n?2:
(6) Dn =
flfl
flfl
flfl
flfl
flfl
fl
1+a1 1 ￠￠￠ 1
1 1+a2 ￠￠￠ 1
...,..,..
1 1 ￠￠￠ 1+an
flfl
flfl
flfl
flfl
flfl
fl
,?a1a2￠￠￠an 6= 0.
3:
6¨E.
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 1 1 ￠￠￠ 1
0 1+a1 1 ￠￠￠ 1
0 1 1+a2 ￠￠￠ 1
...,..,..,..
0 1 1 ￠￠￠ 1+an
flfl
flfl
flfl
flfl
flfl
flfl
fl
(n+1)￡(n+1)
ri?r1=======
i=2;3;￠￠￠
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 1 1 ￠￠￠ 1
1 a1 0 ￠￠￠ 0
1 0 a2 ￠￠￠ 0
...,..,..,..
1 0 0 ￠￠￠ an
flfl
flfl
flfl
flfl
flfl
flfl
fl
(n+1)￡(n+1)
c1+ 1a1c2=======
flfl
flfl
flfl
flfl
flfl
flfl
fl
1+ 1a1 1 1 ￠￠￠ 1
0 a1 0 ￠￠￠ 0
1 0 a2 ￠￠￠ 0
...,..,..,..
1 0 0 ￠￠￠ an
flfl
flfl
flfl
flfl
flfl
flfl
fl
(n+1)￡(n+1)
c1+ 1aj cj+1
=========j=2;3;￠￠￠
flfl
flfl
flfl
flfl
flfl
flfl
fl
1+ 1a1 + 1a2 +￠￠￠+ 1an 1 1 ￠￠￠ 1
0 a1 0 ￠￠￠ 0
0 0 a2 ￠￠￠ 0
...,..,..,..
0 0 0 ￠￠￠ an
flfl
flfl
flfl
flfl
flfl
flfl
fl
(n+1)￡(n+1)
=(a1a2￠￠￠an)
1+
nX
i=1
1
ai
·
:
14?Bc?

T
8,¨ X ?
E53/
Z?F:
(1)
8
>>>>
<
>>>>
:
x1 +x2 +x3 +x4 = 5;
x1 +2x2?x3 +4x4 =?2;
2x1?3x2?x3?5x4 =?2;
3x1 +x2 +2x3 +11x4 = 0;
3:
D =
flfl
flfl
flfl
flfl
flfl
1 1 1 1
1 2?1 4
2?3?1?5
3 1 2 11
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
flfl
1 1 1 1
0 1?2 3
0?5?3?7
0?2?1 8
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
flfl
1 1 1 1
0 1?2 3
0 0?13 8
0 0?5 14
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
flfl
1 1 1 1
0 1?2 3
0 0?1?54
0 0 0 142
flfl
flfl
flfl
flfl
flfl
=?142:
D1 =
flfl
flfl
flfl
flfl
flfl
5 1 1 1
2 2?1 4
2?3?1?5
0 1 2 11
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
flfl
5 1?1?10
0 5?5?18
2?3 5 28
0 1 0 0
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
5?1?10
0?5?18
2 5 28
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
5?1?10
4 0 10
23 0?22
flfl
flfl
flfl
flfl
=?142;
D2 =
flfl
flfl
flfl
flfl
flfl
1 5 1 1
1?2?1 4
2?2?1?5
3 0 2 11
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
flfl
1 5 1 1
0?7?2 3
0?12?3?7
0?15?1 8
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
flfl
1 5 1 1
0?1 3 2
0 0 23 11
0 0 39 31
flfl
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
flfl
1 5 1 1
0?1 3 2
0 0?1?19
0 0 0?284
flfl
flfl
flfl
flfl
flfl
=?284;
D3 =
flfl
flfl
flfl
flfl
flfl
1 1 5 1
1 2?2 4
2?3?2?5
3 1 0 11
flfl
flfl
flfl
flfl
flfl
=?426; D4 =
flfl
flfl
flfl
flfl
flfl
1 1 1 5
1 2?1?2
2?3?1?2
3 1 2 0
flfl
flfl
flfl
flfl
flfl
= 142:
[,
x1 = D1D = 1 ; x2 = D2D = 2 ; x3 = D3D = 3 ; x4 = D4D =?1:
(2)
8
>>>>
>>>
<
>>>>
>>>:
5x1+6x2 = 1;
x1+5x2+6x3 = 0;
x2+5x3+6x4 = 0;
x3+5x4+6x5= 0;
x4+5x5= 1:
3:"
?

T
D5￡5 =
flfl
flfl
flfl
flfl
flfl
flfl
fl
5 6
1 5 6
1 5 6
1 5 6
1 5
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c1====== 5D
4￡4?
flfl
flfl
flfl
flfl
flfl
6 0 0 0
1 5 6 0
0 1 5 6
0 0 1 5
flfl
flfl
flfl
flfl
flfl
= 5D4￡4?6
flfl
flfl
flfl
flfl
5 6 0
1 5 6
0 1 5
flfl
flfl
flfl
flfl
= 5D4￡4?6D3￡3:
??B
TD5￡5 = 5D4￡4?6D3￡3?,
D3￡3 = 5D2￡2?6D1￡1 = 5(25?6)?6￡5 = 65;
D4￡4 = 5D3￡3?6D2￡2 = 5￡65?6￡19 = 211;
L?}
(]6
1?)5? Is?15
D5￡5 = 5D4￡4?6D3￡3 = 5￡211?6￡65 = 665:
?
D1 =
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 6 0 0 0
0 5 6 0 0
0 1 5 6 0
0 0 1 5 6
1 0 0 1 5
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c1====== D
4￡4 +
flfl
flfl
flfl
flfl
flfl
6 0 0 0
5 6 0 0
1 5 6 0
0 1 5 6
flfl
flfl
flfl
flfl
flfl
= D4￡4 +64 = 211+64 = 1507:
D2 =
flfl
flfl
flfl
flfl
flfl
flfl
fl
5 1 0 0 0
1 0 6 0 0
0 0 5 6 0
0 0 1 5 6
0 1 0 1 5
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c2======
flfl
flfl
flfl
flfl
flfl
1 6 0 0
0 5 6 0
0 1 5 6
0 0 1 5
flfl
flfl
flfl
flfl
flfl
+(?1)5+2
flfl
flfl
flfl
flfl
flfl
5 0 0 0
1 6 0 0
0 5 6 0
0 1 5 6
flfl
flfl
flfl
flfl
flfl
Z 7c2======?D
3￡3?5￡63 =?65?1080 =?1145:
D3 =
flfl
flfl
flfl
flfl
flfl
flfl
fl
5 6 1 0 0
1 5 0 0 0
0 1 0 6 0
0 0 0 5 6
0 0 1 1 5
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c3====== (?1)1+3
flfl
flfl
flfl
flfl
flfl
1 5 0 0
0 1 6 0
0 0 5 6
0 0 1 5
flfl
flfl
flfl
flfl
flfl
+(?1)5+3
flfl
flfl
flfl
flfl
flfl
5 6 0 0
1 5 0 0
0 1 6 0
0 0 5 6
flfl
flfl
flfl
flfl
flfl
Z 7c2====== D
2￡2?6￡6￡D2￡2 = 703:
D4 =
flfl
flfl
flfl
flfl
flfl
flfl
fl
5 6 0 1 0
1 5 6 0 0
0 1 5 0 0
0 0 1 0 6
0 0 0 1 5
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c4====== (?1)1+4
flfl
flfl
flfl
flfl
flfl
1 5 6 0
0 1 5 0
0 0 1 6
0 0 0 5
flfl
flfl
flfl
flfl
flfl
+(?1)5+4
flfl
flfl
flfl
flfl
flfl
5 6 0 0
1 5 6 0
0 1 5 0
0 0 1 6
flfl
flfl
flfl
flfl
flfl
Z 7c4======?5?6D
3￡3 =?5?6￡65 =?395:
D5 =
flfl
flfl
flfl
flfl
flfl
flfl
fl
5 6 0 0 1
1 5 6 0 0
0 1 5 6 0
0 0 1 5 0
0 0 0 1 1
flfl
flfl
flfl
flfl
flfl
flfl
fl
Z 7c5====== (?1)1+5
flfl
flfl
flfl
flfl
flfl
1 5 6 0
0 1 5 6
0 0 1 5
0 0 0 1
flfl
flfl
flfl
flfl
flfl
+
flfl
flfl
flfl
flfl
flfl
5 6 0 0
1 5 6 0
0 1 5 6
0 0 1 5
flfl
flfl
flfl
flfl
flfl
= 1+D4￡4 = 1+211 = 212:
[,
x1 = 1507665 ; x2 =?1145665 ; x3 = 703665; x4 =?395665; x5 = 212665:
9,ù?,?|′
H,
QL?Z?F
8
>><
>>:
x1 +x2 +x3 = 0;
x1 +?x2 +x3 = 0;
x1 +2?x2 +x3 = 0:
μd
,3?
16?Bc?

T
3:"
?
D =
flfl
flfl
flfl
flfl
1 1
1? 1
1 2? 1
flfl
flfl
flfl
flfl
r3?r1=====
flfl
flfl
flfl
flfl
1 1
1? 1
0? 0
flfl
flfl
flfl
flfl
=?(?1)3+2
flfl
flfl
fl
1
1 1
flfl
flfl
fl =?(1):
1
P
QL?Z?Fμd
,3,5D = 0,'
(1) = 0;
¤? = 0 = 1.
10,ù?|′
H,
QL?Z?F
8>
><
>>:
(1)x1?2x2 +4x3= 0;
2x1+(3)x2 +x3= 0;
x1 +x2+(1)x3= 0:
μd
,3?
3:"
?
D =
flfl
flfl
flfl
flfl
12 4
2 3 1
1 1 1
flfl
flfl
flfl
flfl
c2?c1=====
flfl
flfl
flfl
flfl
13+? 4
2 1 1
1 0 1
flfl
flfl
flfl
flfl
= 1￡(?1)3+1
flfl
flfl
fl
3+? 4
1 1
flfl
flfl
fl+(1)(?1)
3+3
flfl
flfl
fl
13+?
2 1
flfl
flfl
fl
= (3)?4(1)+(1)3?2(1)(?3+?)
= (1)3 +2(1)2 +3
=?(2)(3):

QL?Z?Fμd
,3,5D = 0,'
(2)(3) = 0:
¤? = 0,? = 2 = 3.
?=c ?#
1,X?L?MD,8
>><
>>:
x1 = 2y1 +2y2 +y3;
x2 = 3y1 +y2 +5y3;
x3 = 3y1 +2y2 +3y3;
pVM
x1;x2;x3?M
y1;y2;y3￥L?MD.
3:ZEB.¨híE3Z?,¤y1,y2,y3.
{.
ZE=.3 ?Z?.?X?:
0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
2 2 1
3 1 5
3 2 3
1
CC
A
0
BB
@
y1
y2
y2
1
CC
A;
#0
BB
@
y1
y2
y2
1
CC
A =
0
BB
@
2 2 1
3 1 5
3 2 3
1
CC
A
10
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
7?4 9
6 3?7
3 2?4
1
CC
A
0
BB
@
y1
y2
y3
1
CC
A:
'8
>><
>>:
y1 =?7x1?4x2 +9x3;
y2 = 6x1 +3x2?7x3;
y3 = 3x1 +2x2?4x3:
ZE?.¨ X ?
E53Z?."
?
D =
flfl
flfl
flfl
flfl
2 2 1
3 1 5
3 2 3
flfl
flfl
flfl
flfl
c1?2c3======
c2?2c3
flfl
flfl
flfl
flfl
0 0 1
7?9 5
3?4 3
flfl
flfl
flfl
flfl
= 1:
[,
y1 = 1D
flfl
flfl
flfl
flfl
x1 2 1
x2 1 5
x3 2 3
flfl
flfl
flfl
flfl
= x1
flfl
flfl
fl
1 5
2 3
flfl
flfl
fl?x2
flfl
flfl
fl
2 1
2 3
flfl
flfl
fl+x3
flfl
flfl
fl
2 1
1 5
flfl
flfl
fl =?7x1?4x2 +9x3 ;
] ?¤y2 = 6x1 +3x2?7x3,y3 = 3x1 +2x2?4x3.
2,X?
?L?MD
8
>><
>>:
x1 = 2y1 +y3;
x2 =?2y1 +3y2 +2y3;
x3 = 4y1 +y2 +5y3;
8
>><
>>:
y1 =?3z1 +z2;
y2 = 2z1 +z3;
y3 =?z2 +3z3;
pVz1,z2,z3?x1,x2,x3￥L?MD.
3:ZEB.°¤}?.1?:
x1 = 2y1 +y3
= 2(?3z1 +z2)+(?z2 +3z3)
=?6z1 +z2 +3z3:
17
18?=c ?#
ZEe?,?á
ì?D V
P¨'cD￥ ??
M.
ZE=.?X?
0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
2 0 1
2 3 2
4 1 5
1
CC
A
0
BB
@
y1
y2
y2
1
CC
A;
0
BB
@
y1
y2
y2
1
CC
A =
0
BB
@
3 1 0
2 0 1
0?1 3
1
CC
A
0
BB
@
z1
z2
z3
1
CC
A:
[,
0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
2 0 1
2 3 2
4 1 5
1
CC
A
0
BB
@
y1
y2
y2
1
CC
A =
0
BB
@
2 0 1
2 3 2
4 1 5
1
CC
A
0
BB
@
3 1 0
2 0 1
0?1 3
1
CC
A
0
BB
@
z1
z2
z3
1
CC
A
=
0
BB
@
6 1 3
12?4 9
10?1 16
1
CC
A
0
BB
@
z1
z2
z3
1
CC
A:
'8
>><
>>:
x1 =?6z1 +z2 +3z3;
x2 = 12z1?4z2 +9z3;
x3 =?10z1?z2 +16z3:
3,
!A =
0
BB
@
1 1 1
1 1?1
1?1 1
1
CC
A,B =
0
BB
@
1 2 3
1?2 4
0 5 1
1
CC
A;
p3AB?2A#ATB:
3:
3AB?2A = 3
0
BB
@
1 1 1
1 1?1
1?1 1
1
CC
A
0
BB
@
1 2 3
1?2 4
0 5 1
1
CC
A?2
0
BB
@
1 1 1
1 1?1
1?1 1
1
CC
A
= 3
0
BB
@
0 5 8
0?5 6
2 9 0
1
CC
A?2
0
BB
@
1 1 1
1 1?1
1?1 1
1
CC
A =
0
BB
@
2 13 22
2?17 20
4 29?2
1
CC
A:
ATB =
0
BB
@
1 1 1
1 1?1
1?1 1
1
CC
A
0
BB
@
1 2 3
1?2 4
0 5 1
1
CC
A =
0
BB
@
0 5 8
0?5 6
2 9 0
1
CC
A:
4,9
/
e:
(1)
0
BB
@
4 3 1
1?2 3
5 7 0
1
CC
A
0
BB
@
7
2
1
1
CC
A; (2) (1; 2; 3)
0
BB
@
3
2
1
1
CC
A;
(3)
0
BB
@
2
1
3
1
CC
A(?1; 2);
L?}
(]6
1?)5? Is?19
(4)
2 1 4 0
1?1 3 4
!
0
BB
BB
@
1 3 1
0?1 2
1?3 1
4 0?2
1
CC
CC
A;
(5) (x1;x2;x3)
0
BB
@
a11 a12 a13
a12 a22 a23
a13 a23 a33
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A;
(6)
0
BB
BB
@
1 2 1 0
0 1 0 1
0 0 2 1
0 0 0 3
1
CC
CC
A
0
BB
BB
@
1 0 3 1
0 1 2?1
0 0?2 3
0 0 0?3
1
CC
CC
A
.
3,(1)
0
BB
@
4 3 1
1?2 3
5 7 0
1
CC
A
0
BB
@
7
2
1
1
CC
A =
0
BB
@
4￡7+3￡2+1￡1
1￡7+(?2)￡2+3￡1
5￡7+7￡2+0￡1
1
CC
A =
0
BB
@
35
6
49
1
CC
A:
(2) (1; 2; 3)
0
BB
@
3
2
1
1
CC
A = (1￡3+2￡2+3￡1) = (10) = 10:
(3)
0
BB
@
2
1
3
1
CC
A(?1; 2) =
0
BB
@
2￡(?1) 2￡2
1￡(?1) 1￡2
3￡(?1) 3￡2
1
CC
A =
0
BB
@
2 4
1 2
3 6
1
CC
A:
(4)
2 1 4 0
1?1 3 4
!
0
BB
BB
@
1 3 1
0?1 2
1?3 1
4 0?2
1
CC
CC
A
=
6?7 8
20?5?6
!
.
(5)
(x1; x2; x3)
0
BB
@
a11 a12 a13
a12 a22 a23
a13 a23 a33
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A
=(a11x1 +a12x2 +a13x3; a12x1 +a22x2 +a23x3; a13x1 +a23x2 +a33x3)
0
BB
@
x1
x2
x3
1
CC
A
=a11x21 +a22x22 +a33x23 +2a12x1x2 +2a13x1x3 +2a23x2x3:
(6)
0
BB
BB
@
1 2 1 0
0 1 0 1
0 0 2 1
0 0 0 3
1
CC
CC
A
0
BB
BB
@
1 0 3 1
0 1 2?1
0 0?2 3
0 0 0?3
1
CC
CC
A
=
0
BB
BB
@
1 2 5 2
0 1 2?4
0 0?4 3
0 0 0?9
1
CC
CC
A
.
5,
!A =
1 2
1 3
!
,B =
1 0
1 2
!
,ù:
(1) AB = BA

(2) (A+B)2 = A2 +2AB +B2

(3) (A+B)(A?B) = A2?B2

3,(1)y1
AB =
3 4
4 6
!; BA =
1 2
3 8
!;
20?=c ?#
[,
AB 6= BA:
(2)y1
(A+B)2 =
2 2
2 5
!?
2 2
2 5
!
=
8 14
14 29
!;
?
A2 +2AB +B2 =
3 8
4 11
!
+
6 8
8 12
!
+
1 0
3 4
!
=
10 16
15 27
!;
#
(A+B)2 6= A2 +2AB +B2:
(3)y1
(A+B)(A?B) =
2 2
2 5
!?
0 2
0 1
!
=
0 6
0 9
!;
7
A2?B2 =
3 8
4 11
!
1 0
3 4
!
=
2 8
1 7
!;
#
(A+B)(A?B) 6= A2?B2:
??,B?e?￥
aE
^,¤?AB 6= BA-a,°¤μ
(A+B)2 = A2 +AB +BA+B2 6= A2 +2AB +B2;
(A+B)(A?B) = A2?AB +BA?B2 6= A2?B2:
6, Q è
a
ü/

5
^p￥:
(1)?A2 = O,5A = O;
(2)?A2 = A,5A = OA = E;
(3)?AX = AY,OA 6= O,5X = Y,
3,(1)|A =
0 1
0 0
!
,A2 = O,?A 6= O.
(2)|A =
1 1
0 0
!
,A2 = A,?A 6= OOA 6= E.
(3)|A =
1 0
0 0
!
,X =
1 1
1 1
!
,Y =
1 1
0 1
!
.
AX = AYOA 6= O?X 6= Y,
5556 A
?e?,
L5
^Q4?á
ì?i ?
?
@?D
p,?
@
,
p,?
@h
?
p.?
^L?}
D?K?^U￥+?p-B,1
?
￥|B°U??p.
á
ì1?i,
ù? ?9μ
ì￥\FE"a\eE",?
^?á
ì
?￥
L
FEaeE
^???]
￥.
￥`?],
￥
=??],??,
￥
p9?].?
^
??]￥)
S? ú￥?]
,
M]￥o?V
^¨
[-￥?ì:|7X,á
ì?1$?B?\M]"7-:=?'é￥?].
???]￥)
S? ú￥\FE"a\eE",μ,C}
D údè<WB?.
7
!A =
1 0
1
!
.pA2;A3;￠￠￠ ;Ak.
L?}
(]6
1?)5? Is?21
3:?9
A2 =
1 0
1
!?
1 0
1
!
=
1 0
2? 1
!;
A3 = A2A =
1 0
2? 1
!?
1 0
1
!
=
1 0
3? 1
!
:
??,An =
1 0
n? 1
!
./¨
DB
,E￡
ü.
?n = 1
H,A?? ?.L
!n = k
H? ?,5n = k +1
H
Ak+1 = AkA =
1 0
k? 1
!?
1 0
1
!
=
1 0
(k +1)? 1
!
:
?
DB
,E?,Ak =
1 0
k? 1
!
.
8
!A =
0
BB
@
1 0
0? 1
0 0?
1
CC
A,pAk.
3:ZEB.
n59
A2 =
0
BB
@
1 0
0? 1
0 0?
1
CC
A
0
BB
@
1 0
0? 1
0 0?
1
CC
A =
0
BB
@
2 2? 1
0?2 2?
0 0?2
1
CC
A;
A3 = A2 ￠A =
0
BB
@
3 3?2 3?
0?3 3?2
0 0?3
1
CC
A:
??,An =
0
BB
@
n n?n?1 n(n?1)2?n?2
0?n n?n?1
0 0?n
1
CC
A; (n > 2):
/¨
DB
,E￡
ü:
?n = 2
H,A?? ?.
L
!n = k
H? ?,5n = k +1
H
Ak+1 = Ak ￠A =
0
BB
@
k k?k?1 k(k?1)2?k?2
0?k k?k?1
0 0?k
1
CC
A
0
BB
@
1 0
0? 1
0 0?
1
CC
A
=
0
BB
@
k+1 (k +1)?k?1 (k+1)k2?k?1
0?k+1 (k +1)?k?1
0 0?k+1
1
CC
A:
?
DB
,E¤￡,Ak =
0
BB
@
k k?k?1 k(k?1)2?k?2
0?k k?k?1
0 0?k
1
CC
A.

￥??
L
^??^¤?￥.? ú
6μB?3E.
22?=c ?#
ZE=.:
A =
0
BB
@
0 0
0? 0
0 0?
1
CC
A+
0
BB
@
0 1 0
0 0 1
0 0 0
1
CC
A,?E +B:
51
An = (?E +B)n
= (?E)n +C1n(?E)n?1B +C2n(?E)n?2B2 +￠￠￠+Bn:
?i?
B =
0
BB
@
0 1 0
0 0 1
0 0 0
1
CC
A; B2 =
0
BB
@
0 0 1
0 0 0
0 0 0
1
CC
A; B3 =
0
BB
@
0 0 0
0 0 0
0 0 0
1
CC
A:
5
Bk =
0
BB
@
0 0 0
0 0 0
0 0 0
1
CC
A; (k > 3):
[
An = (?E +B)n
= (?E)n +C1n(?E)n?1B +C2n(?E)n?2B2
=
0
BB
@
n 0 0
0?n 0
0 0?n
1
CC
A+
0
BB
@
0 n?n?1 0
0 0 n?n?1
0 0 0
1
CC
A+
0
BB
@
0 0 C2n?n?2
0 0 0
0 0 0
1
CC
A
=
0
BB
@
n n?n?1 n(n?1)2?n?2
0?n n?n?1
0 0?n
1
CC
A:
9
!A,B1n¨ ?,OA1? ?,￡
üBTAB9
^? ?.
￡
ü:'1￡(BTAB)T = BTAB.
X?AT = A,?
T(AB)T = BTAT?
(BTAB)T =?(BTA)B￠T
= BT(BTA)T
= BTATB
= BTAB:
¤￡BTAB
^??.
10
!A,B?
^n¨? ?,￡
üAB
^? ?￥ sA1Hq
^AB = BA.
￡
ü:X?AT = A,BT = B,1￡
(AB)T = AB () AB = BA:
1?iB?￥ ?A,B,i?
(A+B)n¤?
d=[
TZ 7
T.1?Ke?￥f?
(A+B)2 = A2 +2AB +B2;
á
ìü???B?? ?."dA,B V?D.7? ú￥?E,B??
^ V?D￥.
L?}
(]6
1?)5? Is?23
s?:?AB = BA.?
(AB)T = BTAT = BA;
[,(AB)T = AB.'AB
^? ?.
A1?,(AB)T = AB.?
(AB)T = BTAT = BA;
[,AB = BA.
11p/
?￥
I ?:
(1)
1 2
2 5
!;
3,(1)?A =
1 2
2 5
!
,jAj = 1 6= 0,
[A V
I.?
A11 = 5; A21 = 2￡(?1); A12 = 2￡(?1); A22 = 1:
¤
A? =
A11 A21
A12 A22
!
=
5?2
2 1
!
A?1 = 1jAjA?
#A?1 = 1jAjA? =
5?2
2 1
!
.
yp:t??P.44 è10￥2
:
a b
c d
!?1
= 1ad?cb
d?b
c a
!
:
?ad?cb 6= 0.
(2)
cos?sin
sin cos
!;
3:
flfl
flfl
fl
cos?sin
sin cos
flfl
flfl
fl = 1,?

2
¤
cos?sin
sin cos
!?1
=
cos sin
sin cos
!
:
(3)
0
BB
@
1 2?1
3 4?2
5?4 1
1
CC
A;
3,jAj = 2,#A?1i.?
A11 =?4; A21 = 2; A31 = 0;
A12 =?13; A22 = 6; A32 =?1;
A13 =?32; A23 = 14; A33 =?2:
#
A?1 = 1jAjA? =
0
BB
@
2 1 0
132 3?12
16 7?1
1
CC
A:
24?=c ?#
(4)
0
BB
BB
B@
a1
a2
...
an
1
CC
CC
CA; (a1a2￠￠￠an 6= 0).
3:?? ?￥?é?
0
BB
BB
B@
a1
a2
...
an
1
CC
CC
CA
1
=
0
BB
BB
B@
1
a1
1
a2
...
1
an
1
CC
CC
CA:
123/
?Z?:
(1)
2 5
1 3
!
X =
4?6
2 1
!;
3,(1) X =
2 5
1 3
!?1?
4?6
2 1
!
=
3?5
1 2
!?
4?6
2 1
!
=
2?23
0 8
!
.
(2) X
0
BB
@
2 1?1
2 1 0
1?1 1
1
CC
A =
1?1 3
4 3 2
!;
3:
X =
1?1 3
4 3 2
!0B
B@
2 1?1
2 1 0
1?1 1
1
CC
A
1
= 13
1?1 3
4 3 2
!0B
B@
1 0 1
2 3?2
3 3 0
1
CC
A
=
2 2 1
83 5?23
!
:
(3)
1 4
1 2
!
X
2 0
1 1
!
=
3 1
0?1
!;
3:
X =
1 4
1 2
!?1?
3 1
0?1
!?
2 0
1 1
!?1
= 112
2?4
1 1
!?
3 1
0?1
!?
1 0
1 2
!
= 112
6 6
3 0
!?
1 0
1 2
!
=
1 1
1
4 0
!
:
(4)
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
AX
0
BB
@
1 0 0
0 0 1
0 1 0
1
CC
A =
0
BB
@
1?4 3
2 0?1
1?2 0
1
CC
A.
3:
X =
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
A
10
BB
@
1?4 3
2 0?1
1?2 0
1
CC
A
0
BB
@
1 0 0
0 0 1
0 1 0
1
CC
A
1
L?}
(]6
1?)5? Is?25
=
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
A
0
BB
@
1?4 3
2 0?1
1?2 0
1
CC
A
0
BB
@
1 0 0
0 0 1
0 1 0
1
CC
A
=
0
BB
@
2?1 0
1 3?4
1 0?2
1
CC
A:
13 ?¨
I ?3/
L?Z?F:
(1)
8>
><
>>:
x1 +2x2 +3x3 = 1;
2x1 +2x2 +5x3 = 2;
3x1 +5x2 +x3 = 3;
3,(1)Z?F VV
U10
BB
@
1 2 3
2 2 5
3 5 1
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
1
2
3
1
CC
A:
#0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
1 2 3
2 2 5
3 5 1
1
CC
A
10
BB
@
1
2
3
1
CC
A =
0
BB
@
1
0
0
1
CC
A:
[8
>><
>>:
x1 = 1;
x2 = 0;
x3 = 0:
(2)
8
>><
>>:
x1?x2?x3 = 2;
2x1?x2?3x3 = 1;
3x1 +2x2?5x3 = 0:
3:Z?F VV
U10
BB
@
1?1?1
2?1?3
3 2?5
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
2
1
0
1
CC
A:
#0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
1?1?1
2?1?3
3 2?5
1
CC
A
10
BB
@
2
1
0
1
CC
A =
0
BB
@
5
0
3
1
CC
A:
[8
>><
>>:
x1 = 5;
x2 = 0;
x3 = 3:
14
!Ak = O (k1??
),￡
ü
(E?A)?1 = E +A+A2 +￠￠￠+Ak?1:
￡
ü:￡
(E?A)(E +A+A2 +￠￠￠+Ak?1) = E
26?=c ?#
' V./
{.
15,
!Z?A
@A2?A?2E = O,￡
üA#A+2E? V
I,ipA?1#(A+2E)?1.
￡
ü:°¤p
I' V.
?A2?A?2E = O
) A(A?E) = 2E
) A
1
2(A?E)
= E
) A?1 = 12(A?E):
??A2?A?2E = O
) (A+2E)A?3(A+2E) =?4E
) (A+2E)(A?3E) =?4E
) (A+2E)
14(A?3E)
= E
) (A+2E)?1 = 14(3E?A):
? ?5
"￥3Eü
^1\X"1p
I￥
T0.1?'5,1VA2?A?2E = O?X
T0A+2E.
16,
!A13¨ ?,jAj = 12,pflfl(2A)?1?5A?flfl.
3:?
jAjflfl(2A)?1?5A?flfl = flflA(2A)?1?5AA?flfl (jABj = jAjjBj)
=
flfl
flfl1
2AA
1?5AA?
flfl
flfl ((?A)?1 = 1?A?1)
=
flfl
flfl1
2E?5jAjE
flfl
flfl
=
flfl
flfl1
2E?
5
2E
flfl
flfl (jAj = 12)
= j?2Ej
= (?2)3jEj (jkAj = knjAj)
=?8:
¤fl
fl(2A)?1?5A?flfl =?16:
17,
! ?A V
I,￡
ü?
?A?9 V
I,O?A?￠?1 =?A?1￠?.
￡
ü:￡AA?1￠? = E' V.
y1?
A?1￠￠?A?1￠? = flflA?1flflE;
7
A?1 = 1jAjA?;'A? = jAjA?1;
[
AA?1￠? = jAjA?1?A?1￠? (A? = jAjA?1)
L?}
(]6
1?)5? Is?27
= jAjflflA?1flflE (?A?1￠￠?A?1￠? = flflA?1flflE)
= E,(jAjflflA?1flfl = flflAA?1flfl = 1)
¤￡ ?A? V
I,O?A?￠?1 =?A?1￠?.
18
!n¨ ?A￥?
?1A?,￡
ü:
(1)?jAj = 0,5jA?j = 0;
(2) jA?j = jAjn?1.
￡
ü,(1)¨Q￡E￡
üL
!jA?j6= 0,5μA?(A?)?1 = E.
?N¤
A = AA?(A?)?1 (A?(A?)?1 = E)
= jAjE(A?)?1 (A(A?) = jAjE)
= O,(jAj = 0)
?DjA?j6= 0
±,#?jAj = 0
HμjA?j = 0.
(2)?AA? = jAjE|?

T¤?:
jAjjA?j = flfljAjEflfl = jAjn,
?jAj6= 05jA?j = jAjn?1.
?jAj = 0?(1)?jA?j = 0N
H
59? ?.
#μ
jA?j = jAjn?1,
19
!A =
0
BB
@
0 3 3
1 1 0
1 2 3
1
CC
A,AB = A+2B,pB.
3:?AB = A+2B¤
(A?2E)B = A:
#
B = (A?2E)?1A =
0
BB
@
2 3 3
1?1 0
1 2 1
1
CC
A
10
BB
@
0 3 3
1 1 0
1 2 3
1
CC
A =
0
BB
@
0 3 3
1 2 3
1 1 0
1
CC
A:
20,
!A =
0
BB
@
1 0 1
0 2 0
1 0 1
1
CC
A,AB +E = A2 +B.pB.
3:?
AB +E = A2 +B
=) AB?B = A2?E
=)?A?E￠B =?A?E￠?A+E￠;
?
jA?Ej =
flfl
flfl
flfl
flfl
0 0 1
0 1 0
1 0 0
flfl
flfl
flfl
flfl
=?1 6= 0;
28?=c ?#
?A?E V
I.
[
B = A+E =
0
BB
@
2 0 1
0 3 0
1 0 2
1
CC
A:
21,
!A = diag(1;?2; 1),A?BA = 2BA?8E.pB.
3:?A = diag(1;?2; 1)?A V
I,O
jAj =?2:
?
A?BA = 2BA?8E
=) A?B = 2B?8A?1 (

T
H·eA?1)
=) jAjB = 2AB?8E (

T
HPeA)
=)?2B = 2AB?8E (jAj =?2)
=)?A+E￠B = 4E;
?
A+E￠ =diag(2;?1; 2);
A+E￠?1 =diag(1
2;?1;
1
2;)
[
B = 4?A+E￠?1 = diag(2;?4; 2):
22,X? ?A￥?
?A? =
0
BB
BB
@
1 0 0 0
0 1 0 0
1 0 1 0
0?3 0 8
1
CC
CC
A
,OABA?1 = BA?1 +3E.pB.
3:3ED
5M
.
ABA?1 = BA?1 +3E
=) AB = B +3A (

T
H·eA)
=) jAjB = A?B +3jAjE (

T
HPeA?)
=)?jAjE?A?￠B = 3jAjE
?
TjA?j = jAjn?1,?9
¤jA?j = 8,
[
jAj3 = 8;'jAj = 2:
¤?
2E?A?￠B = 6E:
?
2E?A? =
0
BB
BB
@
1 0 0 0
0 1 0 0
1 0 1 0
0 3 0?6
1
CC
CC
A;
L?}
(]6
1?)5? Is?29
2E?A?￠?1 =
0
BB
BB
@
1 0 0 0
0 1 0 0
1 0 1 0
0 12 0?16
1
CC
CC
A;
[
B = 6?2E?A?￠?1 =
0
BB
BB
@
6 0 0 0
0 6 0 0
6 0 6 0
0 3 0?1
1
CC
CC
A
:
23
!P?1AP =?,?P =
1?4
1 1
!
,? =
1 0
0 2
!
,pA11.
3,P?1AP =?#A = P?P?1.
[
A11 = P?11P?1:
?
jPj = 3; P?1 = 13
1 4
1?1
!
:
[
A11 =
1?4
1 1
!?
1 0
0 211
!? 1
3
4
3
13?13
!
=
2731 2732
683?684
!
:
24,
!AP = P?,?P =
0
BB
@
1 1 1
1 0?2
1?1 1
1
CC
A,? =
0
BB
@
1
1
5
1
CC
A,p’(A) = A8(5E?6A+A2).
3:?5
!¤A = P?P?1,?i1??,5Ak = P?kP?1.?
’(A) = 5A8?6A9 +A10
= P?5?8?6?9 +?10￠P?1:
= diag(?1; 1; 5),5
8 = diag(1; 1; 58);
9 = diag(?1; 1; 59);
10 = diag(1; 1; 510);
5?8?6?9 +?10 = diag(12; 0; 0):
25,
! ?A,B#A+B? V
I,￡
üA?1 +B?19 V
I,ip
I?.
￡
ü:?A,B#A+B? V
I,AA?1 = A?1A = E,BB?1 = B?1B = E,¤
A?1 +B?1 = A?1E +EB?1
= A?1BB?1 +A?1AB?1
= A?1(B +A)B?1
= A?1(A+B)B?1:
30?=c ?#
[
(A?1 +B?1)?1 = ￡A?1(A+B)B?11
= (B?1)?1(A+B)?1(A?1)?1
= B(A+B)?1A:
26,9
0
BB
BB
@
1 2 1 0
0 1 0 1
0 0 2 1
0 0 0 3
1
CC
CC
A
0
BB
BB
@
1 0 3 1
0 1 2?1
0 0?2 3
0 0 0?3
1
CC
CC
A
.
3,0
BB
BB
@
1 2 1 0
0 1 0 1
0 0 2 1
0 0 0 3
1
CC
CC
A
0
BB
BB
@
1 0 3 1
0 1 2?1
0 0?2 3
0 0 0?3
1
CC
CC
A
,
A1 E
O A2
!?
E B1
O B2
!
=
A1 A1B1 +B2
O A2B2
!
=
0
BB
BB
@
1 2 5 2
0 1 2?4
0 0?4 3
0 0 0?9
1
CC
CC
A
:
27|A = B =?C = D =
1 0
0 1
!
,￡
flfl
flfl
fl
A B
C D
flfl
flfl
fl6=
flfl
flfl
fl
jAj jBj
jCj jDj
flfl
flfl
fl:
3:y1
flfl
flfl
fl
A B
C D
flfl
flfl
fl =
flfl
flfl
flfl
flfl
flfl
1 0 1 0
0 1 0 1
1 0 1 0
0?1 0 1
flfl
flfl
flfl
flfl
flfl
r1?r3=====
r2?r4
flfl
flfl
flfl
flfl
flfl
2 0 0 0
0 2 0 0
1 0 1 0
0?1 0 1
flfl
flfl
flfl
flfl
flfl
= 4;
7flfl
flfl
fl
jAj jBj
jCj jDj
flfl
flfl
fl =
flfl
flfl
fl
1 1
1 1
flfl
flfl
fl = 0;
#flfl
flfl
fl
A B
C D
flfl
flfl
fl6=
flfl
flfl
fl
jAj jBj
jCj jDj
flfl
flfl
fl:
28
!A =
0
BB
BB
@
3 4
4?3 O
O 2 02 2
1
CC
CC
A
,pflflA8flfl#A4.
3::A1 =
3 4
4?3
!
,A2 =
2 0
2 2
!
.
L?}
(]6
1?)5? Is?31
5
A =
A1 O
O A2
!
:
[
A8 =
A1 O
O A2
!8
=
A81 O
O A82
!
:
¤
flflA8flfl = flflA8
1
flflflflA8
2
flfl = jA
1j
8jA
2j
8
=
flfl
flfl
fl
3 4
4?3
flfl
flfl
fl
8flfl
flfl
fl
2 0
2 2
flfl
flfl
fl
8
= 1016:
A4 =
A41 O
O A42
!
=
0
BB
BB
@
54 0
0 54 O
O 2
4 0
26 24
1
CC
CC
A
:
29
!n¨ ?A#s¨ ?B? V
I,p
(1)
O A
B O
!?1;
3:
!?
O An￡n
Bs￡s O
!?
C1 C2
C3 C4
!
= E =
En O
O Es
!;
?C11s￡n ?,C21s￡s ?,C31n￡n ?,C41n￡s ?.?
O An￡n
Bs￡s O
!?
C1 C2
C3 C4
!
=
AC3 AC4
BC1 BC2
!
:
?N¤?8
>>>
><
>>>
>:
AC3 = En ) C3 = A?1;
AC4 = O ) C4 = O; (A?1 V
I)
BC1 = O ) C1 = O; (B?1 V
I)
BC2 = Es ) C2 = B?1:
#?
O A
B O
!?1
=
C1 C2
C3 C4
!
=
O B?1
A?1 O
!
:
(2)
A O
C B
!?1;
3:
!?
A O
C B
!?
A1 A2
A3 A4
!
= E:

A O
C B
!?
A1 A2
A3 A4
!
=
AA1 AA2
CA1 +BA3 CA2 +BA4
!
:
32?=c ?#
¤8
>>>
><
>>>>
:
AA1 = E ) A1 = A?1;
AA2 = O ) A2 = O;
CA1 +BA3 = O ) A3 =?B?1CA?1;
CA2 +BA4 = E ) A4 = B?1:
#?
O A
B O
!?1
=
A1 A2
A3 A4
!
=
A?1 O
B?1CA?1 B?1
!
:
30p/
?￥
I ?:
(1)
0
BB
BB
@
5 2 0 0
2 1 0 0
0 0 8 3
0 0 5 2
1
CC
CC
A;
3:?s v??￥?é?
0
BB
BB
@
5 2
2 1
8 3
5 2
1
CC
CC
A
1
=
0
BB
BB
B@
5 2
2 1
!?1
8 3
5 2
!?1
1
CC
CC
CA
=
0
BB
BB
@
1?2
2 5
2?3
5 8
1
CC
CC
A
:
(2)
0
BB
BB
@
1 0 0 0
1 2 0 0
2 1 3 0
1 2 1 4
1
CC
CC
A
.
3::A =
1 0
1 2
!
,B =
3 0
1 4
!
,C =
2 1
1 2
!
.?
A?1 = 12
2 0
1 1
!; B?1 = 112
4 0
1 3
!;
[0
BB
BB
@
1 0 0 0
1 2 0 0
2 1 3 0
1 2 1 4
1
CC
CC
A
1
=
A?1 O
B?1CA?1 B?1
!
= 124
0
BB
BB
@
24 0 0 0
12 12 0 0
12?4 8 0
3?5?2 6
1
CC
CC
A
:
??c ?￥?MDDL?Z?F
'c￥×15?μ
?:
3 ?Z?￥?ZE,54,5. è5n??P.65 è3.
Z?F3￥)
,515,16,17. è5n??P.76 è12.
1,ü/
?1?Ke? ?:
(1)
0
BB
@
1 0 2?1
2 0 3 1
3 0 4?3
1
CC
A; (2)
0
BB
@
0 2?3 1
0 3?4 3
0 4?7?1
1
CC
A;
(3)
0
BB
BB
@
1?1 3?4 3
3?3 5?4 1
2?2 3?2 0
3?3 4?2?1
1
CC
CC
A; (4)
0
BB
BB
@
2 3 1?3?7
1 2 0?2?4
3?2 8 3 0
2?3 7 4 3
1
CC
CC
A
.
3,(1) 0
BB
@
1 0 2?1
2 0 3 1
3 0 4?3
1
CC
A
r2?2r1
^r
3?3r1
0
BB
@
1 0 2?1
0 0?1 3
0 0?2 0
1
CC
A
r2 ￥(?1)
^r
3 ￥(?2)
0
BB
@
1 0 2?1
0 0 1?3
0 0 1 0
1
CC
A
r3?r2
^
0
BB
@
1 0 2?1
0 0 1?3
0 0 0 3
1
CC
A
r3 ￥3
^
0
BB
@
1 0 2?1
0 0 1?3
0 0 0 1
1
CC
A
r2 + 3r3
^
0
BB
@
1 0 2?1
0 0 1 0
0 0 0 1
1
CC
A
r1?2r2
^r
1 + r3
0
BB
@
1 0 0 0
0 0 1 0
0 0 0 1
1
CC
A
(2) 0
BB
@
0 2?3 1
0 3?4 3
0 4?7?1
1
CC
A
r2 ￡2?3r1
^r
3?2r1
0
BB
@
0 2?3 1
0 0 1 3
0 0?1?3
1
CC
A
r3 + r2
^r
1 + 3r2
0
BB
@
0 2 0 10
0 0 1 3
0 0 0 0
1
CC
A
r1 ￥2
^
0
BB
@
0 1 0 5
0 0 1 3
0 0 0 0
1
CC
A
(3) 0
BB
BB
@
1?1 3?4 3
3?3 5?4 1
2?2 3?2 0
3?3 4?2?1
1
CC
CC
A
r2?3r1
^r3?2r1
r4?3r1
0
BB
BB
@
1?1 3?4 3
0 0?4 8?8
0 0?3 6?6
0 0?5 10?10
1
CC
CC
A
r2 ￥(?4)
^r3 ￥(?3)
r4 ￥(?5)
0
BB
BB
@
1?1 3?4 3
0 0 1?2 2
0 0 1?2 2
0 0 1?2 2
1
CC
CC
A
r1?3r2
^r3?r2
r4?r2
0
BB
BB
@
1?1 0 2?3
0 0 1?2 2
0 0 0 0 0
0 0 0 0 0
1
CC
CC
A
(4) 0
BB
BB
@
2 3 1?3?7
1 2 0?2?4
3?2 8 3 0
2?3 7 4 3
1
CC
CC
A
r1?2r2
^r3?3r2
r4?2r2
0
BB
BB
@
0?1 1 1 1
1 2 0?2?4
0?8 8 9 12
0?7 7 8 11
1
CC
CC
A
33
34??c ?￥?MDDL?Z?F
r2 + 2r1
^r3?8r1
r4?7r1
0
BB
BB
@
0?1 1 1 1
1 0 2 0?2
0 0 0 1 4
0 0 0 1 4
1
CC
CC
A
r1 $ r2
^r2 ￡(?1)
r4?r3
0
BB
BB
@
1 0 2 0?2
0 1?1?1?1
0 0 0 1 4
0 0 0 0 0
1
CC
CC
A
r2 + r3
^
0
BB
BB
@
1 0 2 0?2
0 1?1 0 3
0 0 0 1 4
0 0 0 0 0
1
CC
CC
A
2,
!
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
AA
0
BB
@
1 0 1
0 1 0
0 0 1
1
CC
A =
0
BB
@
1 2 3
4 5 6
7 8 9
1
CC
A,pA.
3: V[?vp ?Z?￥B?ZEp3.
A =
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
A
10
BB
@
1 2 3
4 5 6
7 8 9
1
CC
A
0
BB
@
1 0 1
0 1 0
0 0 1
1
CC
A
1
?0
BB
@
0 1 0 1 2 3
1 0 0 4 5 6
0 0 1 7 8 9
1
CC
A
r1 $ r2
^
0
BB
@
1 0 0 4 5 6
0 1 0 1 2 3
0 0 1 7 8 9
1
CC
A
0
BB
BB
BB
BB
B@
1 0 1
0 1 0
0 0 1
4 5 6
1 2 3
7 8 9
1
CC
CC
CC
CC
CA
c3?c1
^
0
BB
BB
BB
BB
B@
1 0 0
0 1 0
0 0 1
4 5 2
1 2 2
7 8 2
1
CC
CC
CC
CC
CA
¤
A =
0
BB
@
4 5 2
1 2 2
7 8 2
1
CC
A
?μ/
￥3E.
:B =
0
BB
@
1 2 3
4 5 6
7 8 9
1
CC
A.?i?A
He[￥
^? ?, V? ?B
^üAé??MDr1 $ r2
c3 +c1¤?￥.
[1¤?A,31|Bé??MDc3?c1r1 $ r2.'
B =
0
BB
@
1 2 3
4 5 6
7 8 9
1
CC
A
c3?c1
^
0
BB
@
1 2 2
4 5 2
7 8 2
1
CC
A
r1 $ r2
^
0
BB
@
4 5 2
1 2 2
7 8 2
1
CC
A = A
μB? AE
^ü
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
A
0
BB
@
1 0 1
0 1 0
0 0 1
1
CC
AM1?ê?,?M??
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
AA
0
BB
@
1 0 1
0 1 0
0 0 1
1
CC
A
L?}
(]6
1?)5? Is?35
HsYPea·e
M?￥? ?,
*
1 ?
0
BB
@
1 2 3
4 5 6
7 8 9
1
CC
A91é?M?￥?MDa
MD,':
0
BB
@
0 1 0
1 0 0
0 0 1
1
CC
AA
0
BB
@
1 0 1
0 1 0
0 0 1
1
CC
A =
0
BB
@
1 2 3
4 5 6
7 8 9
1
CC
A
=)
0
BB
@
1 0 0
0 1 0
0 0 1
1
CC
AA
0
BB
@
1 0 1
0 1 0
0 0 1
1
CC
A =
0
BB
@
4 5 6
1 2 3
7 8 9
1
CC
A (
H]
Hr1 $ r2)
=)
0
BB
@
1 0 0
0 1 0
0 0 1
1
CC
AA
0
BB
@
1 0 0
0 1 0
0 0 1
1
CC
A =
0
BB
@
4 5 2
1 2 2
7 8 2
1
CC
A (
H]
Hc3?c1)
=)A =
0
BB
@
4 5 2
1 2 2
7 8 2
1
CC
A:
3,
k ?¨ ?￥?MD,p/
Z?￥
I ?:
(1)
0
BB
@
3 2 1
3 1 5
3 2 3
1
CC
A; (2)
0
BB
BB
@
3?2 0?1
0 2 2 1
1?2?3?2
0 1 2 1
1
CC
CC
A
.
3,(1) 0
BB
@
3 2 1 1 0 0
3 1 5 0 1 0
3 2 3 0 0 1
1
CC
A
r2?r1
^r
3?r1
0
BB
@
3 2 1 1 0 0
0?1 4?1 1 0
0 0 2?1 0 1
1
CC
A
r1?r3=2
^r
2?2r3
0
BB
@
3 2 0 32 0?12
0?1 0 1 1?2
0 0 2?1 0 1
1
CC
A
r1?2r2
^r
3 ￥2
0
BB
@
3 0 0 72 2?92
0?1 0 1 1?2
0 0 1?12 0 12
1
CC
A
r1 ￥3
^
0
BB
@
1 0 0 76 23?32
0 1 0?1?1 2
0 0 1?12 0 12
1
CC
A
'
I ?1
0
BB
@
7
6
2
3?
3
2
1?1 2
12 0 12
1
CC
A.
(2) 0
BB
BB
@
3?2 0?1 1 0 0 0
0 2 2 1 0 1 0 0
1?2?3?2 0 0 1 0
0 1 2 1 0 0 0 1
1
CC
CC
A
r1 $ r3
^
0
BB
BB
@
1?2?3?2 0 0 1 0
0 2 2 1 0 1 0 0
3?2 0?1 1 0 0 0
0 1 2 1 0 0 0 1
1
CC
CC
A
r3?3r1
^
0
BB
BB
@
1?2?3?2 0 0 1 0
0 2 2 1 0 1 0 0
0 4 9 5 1 0?3 0
0 1 2 1 0 0 0 1
1
CC
CC
A
r2 $ r4
^
0
BB
BB
@
1?2?3?2 0 0 1 0
0 1 2 1 0 0 0 1
0 4 9 5 1 0?3 0
0 2 2 1 0 1 0 0
1
CC
CC
A
36??c ?￥?MDDL?Z?F
r3?4r1
^r
4?2r1
0
BB
BB
@
1?2?3?2 0 0 1 0
0 1 2 1 0 0 0 1
0 0 1 1 1 0?3?4
0 0?2?1 0 1 0?2
1
CC
CC
A
r4 + 2r3
^
0
BB
BB
@
1?2?3?2 0 0 1 0
0 1 2 1 0 0 0 1
0 0 1 1 1 0?3?4
0 0 0 1 2 1?6?10
1
CC
CC
A
r3?r4
^r2?r4
r1 + 2r4
0
BB
BB
@
1?2?3 0 4 2?11?20
0 1 2 0?2?1 6 11
0 0 1 0?1?1 3 6
0 0 0 1 2 1?6?10
1
CC
CC
A
r2?2r3
^r
1 + 3r3
0
BB
BB
@
1?2 0 0 1?1?2?2
0 1 0 0 0 1 0?1
0 0 1 0?1?1 3 6
0 0 0 1 2 1?6?10
1
CC
CC
A
r1 + 2r2
^
0
BB
BB
@
1 0 0 0 1 1?2?4
0 1 0 0 0 1 0?1
0 0 1 0?1?1 3 6
0 0 0 1 2 1?6?10
1
CC
CC
A
#
I ?1
0
BB
BB
@
1 1?2?4
0 1 0?1
1?1 3 6
2 1?6?10
1
CC
CC
A
.
4,(1)
!A =
0
BB
@
4 1?2
2 2 1
3 1?1
1
CC
A,B =
0
BB
@
1?3
2 2
3?1
1
CC
A,pX
PAX = B;
(2)
!A =
0
BB
@
0 2 1
2?1 3
3 3?4
1
CC
A,B =
1 2 3
2?3 1
!
,pX
PXA = B.
3,(1)
(A; B) =
0
BB
@
4 1?2 1?3
2 2 1 2 2
3 1?1 3?1
1
CC
A
r1?r3
^
0
BB
@
1 0?1?2?2
2 2 1 2 2
3 1?1 3?1
1
CC
A
r2?2r1
^r
3?3r1
0
BB
@
1 0?1?2?2
0 2 3 6 6
0 1 2 9 5
1
CC
A
r2 $ r3
^
0
BB
@
1 0?1?2?2
0 1 2 9 5
0 2 3 6 6
1
CC
A
r3?2r2
^
0
BB
@
1 0?1?2?2
0 1 2 9 5
0 0?1?12?4
1
CC
A
r1?r3
^r2 + 2r3
r3 ￡(?1)
0
BB
@
1 0 0 10 2
0 1 0?15?3
0 0 1 12 4
1
CC
A:
[
X = A?1B =
0
BB
@
10 2
15?3
12 4
1
CC
A:
(2)
A
B
!
=
0
BB
BB
BB
B@
0 2 1
2?1 3
3 3?4
1 2 3
2?3 1
1
CC
CC
CC
CA
c1 $ c3
^
0
BB
BB
BB
B@
1 2 0
3?1 2
4 3?3
3 2 1
1?3 2
1
CC
CC
CC
CA
c2?2c1
^
0
BB
BB
BB
B@
1 0 0
3?7 2
4 11?3
3?4 1
1?5 2
1
CC
CC
CC
CA
L?}
(]6
1?)5? Is?37
c2 + 4c3
^
0
BB
BB
BB
B@
1 0 0
3 1 2
4?1?3
3 0 1
1 3 2
1
CC
CC
CC
CA
c3?2C2
^
0
BB
BB
BB
B@
1 0 0
3 1 0
4?1?1
3 0 1
1 3?4
1
CC
CC
CC
CA
c2?c3
^c1?4c3
c3 ￡(?1)
0
BB
BB
BB
B@
1 0 0
3 1 0
0 0?1
1?1 1
17 7?4
1
CC
CC
CC
CA
c1?3c2
^
0
BB
BB
BB
B@
1 0 0
0 1 0
0 0 1
2?1?1
4 7 4
1
CC
CC
CC
CA
:
[
X = BA?1 =
2?1?1
4 7 4
!
:
'53ED??P.65 è3M],
^3 ?Z?￥B?.×.1.Z.E,9
^B?A?g?￥5?.
35?μB)%?1?i:?135￥ 7
Sü

X?AX = B;
[X = A?1B:
y1A
^? V
I,N
H
^??￥.?ì1?ü?
C?E.
5,
!A =
0
BB
@
1?1 0
0 1?1
1 0 1
1
CC
A,AX = 2X +A,pX.
3:?AX = 2X +A¤
AX?2X = A
(A?2E)X = A
?
(A?2E; A) =
0
BB
@
1?1 0 1?1 0
0?1?1 0 1?1
1 0?1?1 0 1
1
CC
A
r3?r1
^
0
BB
@
1?1 0 1?1 0
0?1?1 0 1?1
0 1?1?2 1 1
1
CC
A
r3 + r2
^
0
BB
@
1?1 0 1?1 0
0?1?1 0 1?1
0 0?2?2 2 0
1
CC
A
r3 ￥(?2)
^
0
BB
@
1?1 0 1?1 0
0?1?1 0 1?1
0 0 1 1?1 0
1
CC
A
r2 + r3
^
0
BB
@
1?1 0 1?1 0
0?1 0 1 0?1
0 0 1 1?1 0
1
CC
A
r2 ￡(?1)
^
0
BB
@
1?1 0 1?1 0
0 1 0?1 0 1
0 0 1 1?1 0
1
CC
A
r1 + r2
^
0
BB
@
1 0 0 0?1 1
0 1 0?1 0 1
0 0 1 1?1 0
1
CC
A
r1 ￡(?1)
^
0
BB
@
1 0 0 0 1?1
0 1 0?1 0 1
0 0 1 1?1 0
1
CC
A
[
X = (A?2E)?1A =
0
BB
@
0 1?1
1 0 1
1?1 0
1
CC
A:
38??c ?￥?MDDL?Z?F
6,?
^r￥ ??,μ
àμ??0￥r?1¨0
T?μ
àμ??0￥r¨0
T?
3:? ??￥?l,?
^r￥ ??,à
iB????0￥r¨0
T,7O
μ¨
ú?r￥
0
T???0.?N V?,?
^r￥ ??, V
i??0￥r?1¨0
T(?
^?
???0),9 V
i??0￥r¨0
T.
è?,A =
0
BB
BB
@
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 0
1
CC
CC
A
,R(A) = 3,?i??0￥3¨0
T2¨0
T.
7,V ?A??B?¤? ?B,ùA,B￥?￥1"8"?
3:?^ü5
!?1:V ?A??::B:::
¤? ?B.
:?￥
*B
1a,5
A? (B; a):
V7,
R(A) = R(B; a):
? ??￥?é5,
R(B) 6 R(B; a) 6 R(B)+1;
[,
R(B) 6 R(A) 6 R(B)+1:
',R(A)??R(B)?R(B)+1.
8,pTB??
^4￥Z?,
￥
??_

^
(1; 0; 1; 0; 0); (1;?1; 0; 0; 0):
3:?Hq￥ ?μ,1?
0
BB
BB
BB
B@
1 0 1 0 0
1?1 0 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0 0 0
1
CC
CC
CC
CA;?
0
BB
BB
BB
B@
1 0 1 0 0
1?1 0 0 0
1 0 0 0 0
0 0 0 1 0
0 0 0 0 0
1
CC
CC
CC
CA
:
?
? ??^sYMD1?¨0?a
¨0? ?,d
,?μ4?.
9,p/
?￥?,ipB?Kú¨d
,0
T:
(1)
0
BB
@
3 1 0 2
1?1 2?1
1 3?4 4
1
CC
A; (2)
0
BB
@
3 2?1?3?1
2?1 3 1?3
7 0 5?1?8
1
CC
A; (3)
0
BB
BB
@
2 1 8 3 7
2?3 0 7?5
3?2 5 8 0
1 0 3 2 0
1
CC
CC
A
.
3,(1) 0
BB
@
3 1 0 2
1?1 2?1
1 3?4 4
1
CC
A
r1 $ r2
^
0
BB
@
1?1 2?1
3 1 0 2
1 3?4 4
1
CC
A
r3?r1
^r
2?3r1
0
BB
@
1?1 2?1
0 4?6 5
0 4?6 5
1
CC
A
r3?r2
^
0
BB
@
1?1 2?1
0 4?6 5
0 0 0 0
1
CC
A:
[ ??12;
L?}
(]6
1?)5? Is?39
=¨0
T
flfl
flfl
fl
3 1
1?1
flfl
flfl
fl =?4,1B?Kú¨d
,0
T.
(2) 0
BB
@
3 2?1?3?2
2?1 3 1?3
7 0 5?1?8
1
CC
A
r1?r2
^
0
BB
@
1 3?4?4 1
2?1 3 1?3
7 0 5?1?8
1
CC
A
r2?2r1
^r
3?7r1
0
BB
@
1 3?4?4 1
0?7 11 9?5
0?21 33 27?15
1
CC
A
r3?3r2
^
0
BB
@
1 3?4?4 1
0?7 11 9?5
0 0 0 0 0
1
CC
A:
[ ??12;
=¨0
T
flfl
flfl
fl
3 2
2?1
flfl
flfl
fl =?7,1B?Kú¨d
,0
T.
(3) 0
BB
BB
@
2 1 8 3 7
2?3 0 7?5
3?2 5 8 0
1 0 3 2 0
1
CC
CC
A
r1?r2
^r
3?r4
0
BB
BB
@
0 4 8?4 12
2?3 0 7?5
2?2 2 6 0
1 0 3 2 0
1
CC
CC
A
r1 ￥4
^r2?r3
r3 ￥2
0
BB
BB
@
0 1 2?1 3
0?1?2 1?5
1?1 1 3 0
1 0 3 2 0
1
CC
CC
A
r3?r4
^
0
BB
BB
@
0 1 2?1 3
0?1?2 1?5
0?1?2 1 0
1 0 3 2 0
1
CC
CC
A
r1 + r3
^r
2?r3
0
BB
BB
@
0 0 0 0 3
0 0 0 0?5
0?1?2 1 0
1 0 3 2 0
1
CC
CC
A
r1 $ r4
^r
2 $ r3
0
BB
BB
@
1 0 3 2 0
0?1?2 1 0
0 0 0 0?5
0 0 0 0 3
1
CC
CC
A
^
0
BB
BB
@
1 0 3 2 0
0?1?2 1 0
0 0 0 0 1
0 0 0 0 0
1
CC
CC
A
:
[ ??13;
?¨0
T
flfl
flfl
flfl
flfl
0 7?5
5 8 0
3 2 0
flfl
flfl
flfl
flfl
=?5
flfl
flfl
fl
5 8
3 2
flfl
flfl
fl = 70 6= 0,1B?Kú¨d
,0
T.
10,
!A,B?
^m￡n ?,￡
üA? B￥ sA1Hq
^R(A) = R(B).
￡
ü:A1?'??P.68? ?3.o3￡
ü s?.
!R(A) = r,5A￥S? ?1F =
Er O
O O
!
,'
A?
Er O
O O
!
:
]",B￥S? ?91F =
Er O
O O
!
,'
B?
Er O
O O
!
:
40??c ?￥?MDDL?Z?F
? ??N￥.??,¤
A? B:
11,
!
A =
0
BB
@
1?2 3k
1 2k?3
k?2 3
1
CC
A;
ùk1′, V
P
(1) R(A) = 1; (2) R(A) = 2; (3) R(A) = 3.
3:
A =
0
BB
@
1?2 3k
1 2k?3
k?2 3
1
CC
A
c2 ￥2
^c
3 ￥3
0
BB
@
1?1 k
1 k?1
k?1 1
1
CC
A
r2 + r1
^r
3?kr1
0
BB
@
1?1 k
0 k?1 k?1
0 k?1 1?k2
1
CC
A^
r3?r2
0
BB
@
1?1 k
0 k?1 k?1
0 0 2?k?k2
1
CC
A;
[,?k 6= 1Ok 6=?2
H,R(A) = 3.
?k = 1
H,A?
0
BB
@
1?1 1
0 0 0
0 0 0
1
CC
A,R(A) = 1.
?k =?2
H,A?
0
BB
@
1?1?2
0?3?3
0 0 0
1
CC
A,R(A) = 2.
63:?jAj6= 0
H,R(A) = 3,'
A =
flfl
flfl
flfl
flfl
1?2 3k
1 2k?3
k?2 3
flfl
flfl
flfl
flfl
= 6
flfl
flfl
flfl
flfl
1?1 k
1 k?1
k?1 1
flfl
flfl
flfl
flfl
r3?kr1======
r2+r1 6
flfl
flfl
flfl
flfl
1?1 k
0 k?1 k?1
0 k?1 1?k2
flfl
flfl
flfl
flfl
r3?r2=====6
flfl
flfl
flfl
flfl
1?1 k
0 k?1 k?1
0 0 2?k?k2
flfl
flfl
flfl
flfl
=?6(k?1)2(k +2):
[,?k 6= 1Ok 6=?2
H,R(A) = 3.
?/￥)
D-M].
12,p3/

QL?Z?F:
(1)
8>
><
>>:
x1 +x2 +2x3?x4 = 0;
2x1 +x2 +x3?x4 = 0;
2x1 +2x2 +x3 +2x4 = 0;
(2)
8>
><
>>:
x1 +2x2 +x3?x4 = 0;
3x1 +6x2?x3?3x4 = 0;
5x1 +10x2 +x3?5x4 = 0;
(3)
8
>>>>
<
>>>
>:
2x1 +3x2?x3 +5x4 = 0;
3x1 +x2 +2x3?7x4 = 0;
4x1 +x2?3x3 +6x4 = 0;
x1?2x2 +4x3?7x4 = 0;
(4)
8
>>>>
<
>>>>
:
3x1 +4x2?5x3 +7x4 = 0;
2x1?3x2 +3x3?2x4 = 0;
4x1 +11x2?13x3 +16x4 = 0;
7x1?2x2 +x3 +3x4 = 0:
L?}
(]6
1?)5? Is?41
3,(1)::"::
:: ::?
L
@?MD:
0
BB
@
1 1 2?1
2 1 1?1
2 2 1 2
1
CC
A
r3?r2
^r
2?r1
0
BB
@
1 1 2?1
1 0?1 0
0 1 0 3
1
CC
A
r1?r2?r3
^
0
BB
@
0 0 3?4
1 0?1 0
0 1 0 3
1
CC
A
r1 ￥3
^
0
BB
@
0 0 1?43
1 0?1 0
0 1 0 3
1
CC
A
r2 + r1
^
0
BB
@
0 0 1?43
1 0 0?43
0 1 0 3
1
CC
A^
0
BB
@
1 0 0?43
0 1 0 3
0 0 1?43
1
CC
A:
¤?eZ?￥?N?
T8
>>>>
<
>>>>
:
x1 = 43x4;
x2 =?3x4;
x3 = 43x4;
x4 = x4:
#Z?F￥31
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
= c
0
BB
BB
@
4
3
3
4
3
1
1
CC
CC
A
,c1?i
L
.
(2)"
?
L
@?MD:
0
BB
@
1 2 1?1
3 6?1?3
5 10 1?5
1
CC
A
r2?3r1
^r
3?5r1
0
BB
@
1 2 1?1
0 0?4 0
0 0?4 0
1
CC
A
r3?r2
^r
2 ￥(?4)
0
BB
@
1 2 1?1
0 0 1 0
0 0 0 0
1
CC
A
r1?r2
^
0
BB
@
1 2 0?1
0 0 1 0
0 0 0 0
1
CC
A:
¤?eZ?￥?N?
T8
>>>>
<
>>>
>:
x1 =?2x2 +x4;
x2 = x2;
x3 = 0;
x4 = x4:
#Z?F￥31
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
= c1
0
BB
BB
@
2
1
0
0
1
CC
CC
A
+c2
0
BB
BB
@
1
0
0
1
1
CC
CC
A
,c1,c21?i
L
.
(3)"
?
L
@?MD:
0
BB
BB
@
2 3?1 5
3 1 2?7
4 1?3 6
1?2 4?7
1
CC
CC
A
r1 $ r4
^
0
BB
BB
@
1?2 4?7
2 3?1 5
3 1 2?7
4 1?3 6
1
CC
CC
A
42??c ?￥?MDDL?Z?F
r2?3r1
^r3?4r1
r2?2r1
0
BB
BB
@
1?2 4?7
0 7?10 14
0 9?19 34
0 7?9 19
1
CC
CC
A
r4?r2
^7r
3?9r2
0
BB
BB
@
1?2 4?7
0 7?10 14
0 0?43 112
0 0 1 5
1
CC
CC
A
r3 + 43r4
^
0
BB
BB
@
1?2 4?7
0 7?10 14
0 0 0 327
0 0 1 5
1
CC
CC
A^
0
BB
BB
@
1?2 4?7
0 7?10 14
0 0 1 5
0 0 0 1
1
CC
CC
A
^
0
BB
BB
@
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1
CC
CC
A
:
[Z?F￥31
8>
>>><
>>>
>:
x1 = 0;
x2 = 0;
x3 = 0;
x4 = 0:
(4)"
?
L
@?MD:
0
BB
BB
@
3 4?5 7
2?3 3?2
4 11?13 16
7?2 1 3
1
CC
CC
A
r3?2r1 + r1
^r
4?r1?2r2
0
BB
BB
@
3 4?5 7
2?3 3?2
0 0 0 0
0 0 0 0
1
CC
CC
A
3r2?2r1
^
0
BB
BB
@
3 4?5 7
0?17 19?20
0 0 0 0
0 0 0 0
1
CC
CC
A^
0
BB
BB
@
1 0? 317 1317
0 1?1917 2017
0 0 0 0
0 0 0 0
1
CC
CC
A
:
'¤
8
>>>>
<
>>>>
:
x1 = 317x3? 1317x4;
x2 = 1917x3? 2017x4;
x3 = x3;
x4 = x4:
[Z?F￥31
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
= c1
0
BB
BB
@
3
17
19
17
1
0
1
CC
CC
A
+c2
0
BB
BB
@
1317
2017
0
1
1
CC
CC
A
,c1,c21?i
L
.
13,p3/
d
QL?Z?F:
(1)
8
>><
>>:
4x1 +2x2?x3 = 2;
3x1?1x2 +2x3 = 10;
11x1 +3x2 = 8;
(2)
8>
>>>
<
>>>>
:
2x+3y +z = 4;
x?2y +4z =?5;
3x+8y?2z = 13;
4x?y +9z =?6;
(3)
8>
><
>>:
2x+y?z +w = 1;
4x+2y?2z +w = 2;
2x+y?z?w = 1;
(4)
8>
><
>>:
2x+y?z +w = 1;
3x?2y +z?3w = 4;
x+4y?3z +5w =?2;
L?}
(]6
1?)5? Is?43
3,(1)"
￥9< ?
L
@?MD,μ
0
BB
@
4 2?1 2
3?1 2 10
11 3 0 8
1
CC
A
r3?2r1?r2
^4r
2?3r1
0
BB
@
1 3?3?8
0?10 11 34
0 0 0?6
1
CC
A;
An
R(A) = 2;7R(B) = 3;
#Z?Fí3.
(2)"
￥9< ?
L
@?MD:
0
BB
BB
@
2 3 1 4
1?2 4?5
3 8?2 13
4?1 9?6
1
CC
CC
A
r3?2r1 + r2
^r
4?r1?2r2
0
BB
BB
@
2 3 1 4
1?2 4?5
0 0 0 0
0 0 0 0
1
CC
CC
A
r1 $ r2
^
0
BB
BB
@
1?2 4?5
2 3 1 4
0 0 0 0
0 0 0 0
1
CC
CC
A
r2?2r1
^
0
BB
BB
@
1?2 4?5
0 7?7 14
0 0 0 0
0 0 0 0
1
CC
CC
A
^
0
BB
BB
@
1 0 2?1
0 1?1 2
0 0 0 0
0 0 0 0
1
CC
CC
A
:
'¤8
>><
>>:
x =?2z?1;
y = z +2;
z = z:
[
0
BB
@
x
y
z
1
CC
A = c
0
BB
@
2
1
1
1
CC
A+
0
BB
@
1
2
0
1
CC
A,c1?i
L
.
(3)"
￥9< ?
L
@?MD:
0
BB
@
2 1?1 1 1
4 2?2 1 2
2 1?1?1 1
1
CC
A
r2?2r1
^r
3?r1
0
BB
@
2 1?1 1 1
0 0 0?1 0
0 0 0?2 0
1
CC
A^
0
BB
@
2 1?1 1 1
0 0 0 1 0
0 0 0 0 0
1
CC
A
'¤8
>>>
><
>>>
>:
x =?12y + 12z + 12;
y = y;
z = z;
w = 0:
[
0
BB
BB
@
x
y
z
w
1
CC
CC
A
= c1
0
BB
BB
@
12
1
0
0
1
CC
CC
A
+c2
0
BB
BB
@
1
2
0
1
0
1
CC
CC
A
+
0
BB
BB
@
1
2
0
0
0
1
CC
CC
A
,c1,c21?i
L
.
44??c ?￥?MDDL?Z?F
(4)"
￥9< ?
L
@?MD:
0
BB
@
2 1?1 1 1
3?2 1?3 4
1 4?3 5?2
1
CC
A
r3?2r1 + r2
^
0
BB
@
2 1?1 1 1
3?2 1?3 4
0 0 0 0 0
1
CC
A
2r2?3r1
^
0
BB
@
2 1?1 1 1
0?7 5?9?5
0 0 0 0 0
1
CC
A^
0
BB
@
1 0?17?17 67
0 1?57 97?57
0 0 0 0 0
1
CC
A
'¤8
>>>>
<
>>>
>:
x = 17z + 17w + 67;
y = 57z? 97w? 57;
z = z;
w = w:
[
0
BB
BB
@
x
y
z
w
1
CC
CC
A
= c1
0
BB
BB
@
1
7
5
7
1
0
1
CC
CC
A
+c2
0
BB
BB
@
1
7
97
0
1
1
CC
CC
A
+
0
BB
BB
@
6
7
57
0
0
1
CC
CC
A
,c1,c21?i
L
.
14,B?[
x = c1
0
BB
BB
@
2
3
1
0
1
CC
CC
A
+c2
0
BB
BB
@
2
4
0
1
1
CC
CC
A
1Y3￥
QL?Z?F.
3:ü3￥?
T?1
8>
>>>
<
>>>>
:
x1 = 2c1?2c2;
x2 =?3c1 +4c2;
x3 = c1;
x4 = c2:
,'
8
>>>>
<
>>>>
:
x1 = 2x3?2x4;
x2 =?3x3 +4x4;
x3 = x3;
x4 = x4:
¤B?
p￥Z?F(
x1?2x3 +2x4 = 0;
x2 +3x3?4x4 = 0:
15,?|′
H,d
QL?Z?F
8
>><
>>:
x1 +x2 +x3 = 1;
x1 +?x2 +x3 =?;
x1 +x2 +?x3 =?2
(1)μ?B3; (2)í3; (3)μík?3?
L?}
(]6
1?)5? Is?45
3,(1):"
?1A,?jAj6= 0
HZ?Fμ?B3.?
flfl
flfl
flfl
flfl
1 1
1? 1
1 1?
flfl
flfl
flfl
flfl
c1+c2+c3========
flfl
flfl
flfl
flfl
+2 1 1
+2? 1
+2 1?
flfl
flfl
flfl
flfl
= (?+2)
flfl
flfl
flfl
flfl
1 1 1
1? 1
1 1?
flfl
flfl
flfl
flfl
r2?r1=====
r3?r1 (?+2)
flfl
flfl
flfl
flfl
1 1 1
01 0
0 01
flfl
flfl
flfl
flfl
= (?+2)(1)2;
¤? 6= 1,O? 6=?2
HZ?Fμ?B3.
(2)? =?2
H,
B =
0
BB
@
2 1 1 1
1?2 1?2
1 1?2 4
1
CC
A
r1 $ r3
^
0
BB
@
1 1?2 4
1?2 1?2
2 1 1 1
1
CC
A
r2?r1
^r
3 + 2r1
0
BB
@
1 1 2 4
0?3 3?6
0 3?3 9
1
CC
A
r3 + r2
^
0
BB
@
1 1 2 4
0?3 3?6
0 0 0 3
1
CC
A:
?R(A) = 2 < R(B) = 3,N
HZ?Fí3.
(3)? = 1
H,
B =
0
BB
@
1 1 1 1
1 1 1 1
1 1 1 1
1
CC
A
r2?r1
^r
3?r1
0
BB
@
1 1 1 1
0 0 0 0
0 0 0 0
1
CC
A:
?R(A) = R(B) = 1,N
HZ?FμíK3.
á
ì<?,?B ?￥5?
^A?g?￥.÷B?￥3E
^,YV?MD,)
R(A)DR(B)￥1".
1??B5,
B =
0
BB
@
1 1 1
1? 1?
1 12
1
CC
A?
0
BB
@
1 12
01 1(1)
0 0 (1)(2+?) (1)(?+1)2
1
CC
A:
?(1)(2+?) = 0,O(1)(1+?)2 6= 0,'? =?2
H,Z?Fí3.
e￥)
?
.
16,d
QL?Z?F8
>><
>>:
2x1 +x2 +x3 =?2;
x1?2x2 +x3 =?;
x1 +x2?2x3 =?2;
|′
Hμ3?ip
￥3.
3:
B =
0
BB
@
2 1 1?2
1?2 1?
1 1?2?2
1
CC
A
r1 $ r2
^
0
BB
@
1?2 1?
2 1 1?2
1 1?2?2
1
CC
A
r2 + 2r1
^r
3?r1
0
BB
@
1?2 1?
0?3 3?2+2?
0 3?3?2
1
CC
A
r3 + r2
^
0
BB
@
1?2 1?
0?3 3?2+2?
0 0 0 (1)(?+2)
1
CC
A:
46??c ?￥?MDDL?Z?F
1
PZ?Fμ3,?(1)(?+2) = 0,¤
= 1; =?2:
= 1
H,Z?F310
BB
@
x1
x2
x3
1
CC
A = c
0
BB
@
1
1
1
1
CC
A+
0
BB
@
1
0
0
1
CC
A:
=?2
H,Z?F310
BB
@
x1
x2
x3
1
CC
A = c
0
BB
@
1
1
1
1
CC
A+
0
BB
@
2
2
0
1
CC
A:
17,
!8
>><
>>:
(2)x1 +2x2?2x3 = 1;
2x1 +(5)x2?4x3 = 2;
2x1?4x2 +(5)x3 =1;
ù?1′
H,NZ?Fμ?B3aí3μík3?iμík3
HpY3.
3:?jAj6= 0
H,μ?B3.
A =
flfl
flfl
flfl
flfl
2 2?2
2 54
2?4 5
flfl
flfl
flfl
flfl
r3+r2=====
flfl
flfl
flfl
flfl
2 2?2
2 54
0 1 1
flfl
flfl
flfl
flfl
= (1)
flfl
flfl
flfl
flfl
2 2?2
2 54
0 1 1
flfl
flfl
flfl
flfl
r2+4r1======
r1+2r3 (1)
flfl
flfl
flfl
flfl
2 4 0
2 9 0
0 1 1
flfl
flfl
flfl
flfl
= (1)
flfl
flfl
fl
2 4
2 9
flfl
flfl
fl = (1)
2(10):
6= 1O? 6= 10
H,μ?B3.
= 10
H,9< ?1
B =
0
BB
@
8 2?2 1
2?5?4 2
2?4?5?11
1
CC
A
r1 + 4r2
^r
3 + r2
0
BB
@
0?18?18 9
2?5?4 2
0?9?9?9
1
CC
A
r1?2r3
^
0
BB
@
0 0 0 27
2?5?4 2
0?9?9?9
1
CC
A
VnR(A) = 2,R(B) = 3,N
HZ?Fí3.
= 1
H,9< ?1
B =
0
BB
@
1 2?2 1
2 4?4 2
2?4 4?2
1
CC
A?
0
BB
@
1 2?2 1
0 0 0 0
0 0 0 0
1
CC
A;
N
HZ?FμíK3.OeZ?F￥31
0
BB
@
x1
x2
x3
1
CC
A = c1
0
BB
@
2
1
0
1
CC
A+c2
0
BB
@
2
0
1
1
CC
A+
0
BB
@
1
0
0
1
CC
A; (c1; c2 2R):
18,￡
üR(A) = 1￥ sA1Hq
^id
,
_
a#d
,?_
bT,
PA = abT.
L?}
(]6
1?)5? Is?47
￡
ü,( s?)
!A = abT.? ??￥?é7μ
R(abT) 6 R(a),(3.1)
?R(a) = 1,
[
R(A) = R(abT) 6 1,(3.2)
?
^R(A) 6= 0,y1d
,_
a,bTsYà
iB?d
,í
í,:1ai,bj.5 ?A?à
μB?
d
,íaibj('A￥B¨d
,0
T),V7
R(A) > 1,(3.3)
8(3.2),(3.3),
[R(A) = 1.
(A1?)
!R(A) = 1.5i V
I ?P,Q,
P¤
A = P
1 O
O O
!
Q = P
0
BB
BB
B@
1
0
...
0
1
CC
CC
CA(1; 0 ￠￠￠ ; 0)Q:
:
a,P
0
BB
BB
B@
1
0
...
0
1
CC
CC
CA; b,(1; 0 ￠￠￠ ; 0)Q;
?i?P,Q
^ V
I ?,?? V
μ?1
,￥?
,V7a,bT'
^
@Hq￥d
,_
.
19,
!A1m￡n ?,￡
ü
(1)Z?AX = Emμ3￥ sA1Hq
^R(A) = m;
(2)Z?YA = Enμ3￥ sA1Hq
^R(A) = n.
￡
ü,(1)???P.78? ?7,Z?AX = Emμ3￥ sA1Hq
^
R(A) = R(A; Em):
A1m￡n ?,5
R(A) 6 m:
7? ??￥?é5,
m = R(Em) 6 R(A; Em) = R(A):
[,R(A) = m.
(2)Z?YA = En
HT?,¤
ATY T = En:
N
HAT1n￡m ?,?(1)￥￡
ü?ATY T = Enμ3￥ sA1Hq
^R(A) = n.V7e
5¤￡.
20,
!A1m￡n ?,￡
ü:?AX = AY,OR(A) = n,5X = Y,
￡
ü:?AX = AY,¤
A(X?Y ) = 0:
?R(A) = n,???P.78? ?9,5 ?Z?A(X?Y ) = 0oμ
,3,'
X?Y · 0:
¤￡
X = Y,
?
1c_
F￥L?M1?
1,
!v1 =
0
BB
@
1
1
0
1
CC
A,v2 =
0
BB
@
0
1
1
1
CC
A,v3 =
0
BB
@
3
4
0
1
CC
A,pv1?v2#3v1 +2v2?v3.
3:
v1?v2 =
0
BB
@
1
1
0
1
CC
A?
0
BB
@
0
1
1
1
CC
A =
0
BB
@
1?0
1?1
0?1
1
CC
A =
0
BB
@
1
0
1
1
CC
A;
3v1 +2v2?v3 = 3
0
BB
@
1
1
0
1
CC
A+2
0
BB
@
0
1
1
1
CC
A?
0
BB
@
3
4
0
1
CC
A =
0
BB
@
3￡1+2￡0?3
3￡1+2￡1?4
3￡0+2￡1?0
1
CC
A =
0
BB
@
0
1
2
1
CC
A:
2,
!3(a1?a)+2(a2 +a) = 5(a3 +a),pa,?
a1 =
0
BB
BB
@
2
5
1
3
1
CC
CC
A; a2 =
0
BB
BB
@
10
1
5
10
1
CC
CC
A; a3 =
0
BB
BB
@
4
1
1
1
1
CC
CC
A
:
3:?3(a1?a)+2(a2 +a) = 5(a3 +a)¤
a = 16(3a1 +2a2?5a3) = 12
0
BB
BB
@
2
5
1
3
1
CC
CC
A
+ 13
0
BB
BB
@
10
1
5
10
1
CC
CC
A
56
0
BB
BB
@
4
1
1
1
1
CC
CC
A
=
0
BB
BB
@
1
2
3
4
1
CC
CC
A
:
3,X?_
F
A,a1 =
0
BB
BB
@
0
1
2
3
1
CC
CC
A; a2 =
0
BB
BB
@
3
0
1
2
1
CC
CC
A; a3 =
0
BB
BB
@
2
3
0
1
1
CC
CC
A; B,b1 =
0
BB
BB
@
2
1
1
2
1
CC
CC
A; b2 =
0
BB
BB
@
0
2
1
1
1
CC
CC
A; b3 =
0
BB
BB
@
4
4
1
3
1
CC
CC
A
:
￡
üBF
?AFL?V
U,?AF?
?BFL?V
U.
￡
ü:1￡BF
?AFL?V
U,?P.85? ?2,'1￡ ?A = (a1; a2; a3)￥??? ?
(A; B) = (a1; a2; a3; b1; b2; b3)￥?.?
(A; B) =
0
BB
BB
@
0 3 2 2 0 4
1 0 3 1?2 4
2 1 0 1 1 1
3 2 1 2 1 3
1
CC
CC
A
r3?2r2
^r
4?3r2
0
BB
BB
@
0 3 2 2 0 4
1 0 3 1?2 4
0 1?6?1 5?7
0 2?8?1 7?9
1
CC
CC
A
r1?3r3
^r
4?2r3
0
BB
BB
@
0 0 20 5 15 25
1 0 3 1?2 4
0 1?6?1 5?7
0 0 4 1 3 5
1
CC
CC
A^
0
BB
BB
@
1 0 3 1?2 4
0 1?6?1 5?7
0 0 4 1 3 5
0 0 0 0 0 0
1
CC
CC
A;
Vn,R(A) = R(A; B) = 3,yN,BF
?AFL?V
U.
48
L?}
(]6
1?)5? Is?49
?
B =
0
BB
BB
@
2 0 4
1?2 4
1 1 1
2 1 3
1
CC
CC
A
r1?2r3
^r2?r3
r4?2r3
0
BB
BB
@
0?2 2
0?3 3
1 1 1
0?1 1
1
CC
CC
A^
0
BB
BB
@
1 1 1
0?1 1
0 0 0
0 0 0
1
CC
CC
A;
VnR(B) = 2.7R(A) = 3,
[AF?
?BFL?V
U,(y1?AF
?BFL?V
U,5
R(A) 6 R(B).n??P.86? ?3.)
4,X?_
F
A,a1 =
0
BB
@
0
1
1
1
CC
A; a2 =
0
BB
@
1
1
0
1
CC
A; B,b1 =
0
BB
@
1
0
1
1
CC
A; b2 =
0
BB
@
1
2
1
1
CC
A; b3 =
0
BB
@
3
2
1
1
CC
A;
￡
üAFDBF?N.
￡
ü:ZED??P.86 è2M]./￡R(A) = R(B) = R(A; B).
(A; B) =
0
BB
@
0 1?1 1 3
1 1 0 2 2
1 0 1 1?1
1
CC
A
r2?r1?r3
^
0
BB
@
0 1?1 1 3
0 0 0 0 0
1 0 1 1?1
1
CC
A^
0
BB
@
1 0 1 1?1
0 1?1 1 3
0 0 0 0 0
1
CC
A;
[R(A) = R(A; B) = 2.??

?MD?
B?
0
BB
@
1 1?1
1 1 3
0 0 0
1
CC
A
r2 + r1
^
0
BB
@
1 1?1
0 2 2
0 0 0
1
CC
A;
VnR(B) = 2.¤?R(A) = R(B) = R(A; B).'￡AFDBF?N.
5,X?R(a1; a2; a3) = 2,R(a2; a3; a4) = 3,￡
ü
(1) a1
?a2; a3L?V
U;
(2) a4?
?a1; a2; a3L?V
U.
￡
ü,(1)?R(a2; a3; a4) = 3,?a2; a3; a4L?í1.V7_
Fa2; a3L?í1(?8í15?s
9í1.nP.90? ?5(1)).
?R(a1; a2; a3) = 2 < 3,
[_
Fa1; a2; a3L?M1.?P.90? ?5(3)?a1
?a2; a3L?V
U.
(2)L
!a4
?a1; a2; a3L?V
U.X￡a1
?a2; a3L?V
U,
[a4
?a2; a3L?V
U.
?DR(a2; a3; a4) = 3
±.
V7¤￡a4?
?a1; a2; a3L?V
U.
6,
?/
_
F
^L?M1
^L?í1:
(1)
0
BB
@
1
3
1
1
CC
A;
0
BB
@
2
1
0
1
CC
A;
0
BB
@
1
4
1
1
CC
A; (2)
0
BB
@
2
3
0
1
CC
A;
0
BB
@
1
4
0
1
CC
A;
0
BB
@
0
0
2
1
CC
A.
3,(1)y1
A =
0
BB
@
1 2 1
3 1 4
1 0 1
1
CC
A
r2 + 3r1
^r
3 + r1
0
BB
@
1 2 1
0 7 7
0 2 2
1
CC
A^
0
BB
@
1 2 1
0 1 1
0 0 0
1
CC
A;
VnR(A) = 2,
[?_
F
^L?M1￥.
50?
1c_
F￥L?M1?
(2)y1
B =
0
BB
@
2?1 0
3 4 0
0 0 2
1
CC
A
r2 ￡2?3r1
^
0
BB
@
2?1 0
0 11 0
0 0 2
1
CC
A;
VnR(B) = 3,
[?_
F
^L?í1￥.
7,ùa|
I
1′
H/
_
FL?M1?
a1 =
0
BB
@
a
1
1
1
CC
A; a2 =
0
BB
@
1
a
1
1
CC
A; a3 =
0
BB
@
1
1
a
1
CC
A:
3:_
Fa1; a2; a3L?M1￥ 1Hq
^R(a1; a2; a3) < 3,'jAj = 0,? úA = (a1; a2; a3).?
jAj =
flfl
flfl
flfl
flfl
a 1 1
1 a?1
1?1 a
flfl
flfl
flfl
flfl
r1?ar3======
r2?r3
flfl
flfl
flfl
flfl
0 a+1 1?a2
0 a+1?1?a
1?1 a
flfl
flfl
flfl
flfl
= (a+1)2(a?2):
[,a =?1a = 2
H_
FL?M1.
8,
!a1; a2L?í1,a1 +b; a2 +bL?M1,p_
b¨a1; a2L?V
U￥V
U
T.
3:
!b = k1a1 +k2a2.?a1 +b; a2 +bL?M1,5i??1
,￥x1,x2
P¤
x1(a1 +b)+x2(a2 +b) = 0:
}?b = k1a1 +k2a2,¤
￡x
1(k1 +1)+x2k1
a
1 +
￡x
1k2 +x2(k2 +1)
a
2 = 0,(4.1)
?a1; a2L?í1,¤(
x1(k1 +1)+x2k1 = 0;
x1k2 +x2(k2 +1) = 0,(4.2)
1
Px1,x2??1
,,5Z?F(4.2)?"
?

T1
,,'
flfl
flfl
fl
k1 +1 k1
k2 k2 +1
flfl
flfl
fl = k1 +k2 +1 = 0:
[
b = k1a1 +k2a2 = k1a1?(1+k1)a2;
?:1
b = ca1?(1+c)a2; (c 2R):
9,
!a1; a2L?í1,b1; b29L?í1,ùa1 +b1; a2 +b2
^?B?L?M1?
k è
a
ü-.
3:?B?.1?|
a1 =
1
0
!; a2 =
2
0
!; b1 =
0
3
!; b2 =
0
4
!;
¤
a1 +b1 =
1
3
!; a2 +b2 =
2
4
!;
L?}
(]6
1?)5? Is?51
N
Ha1; a2L?í1,b1; b29L?í1,?
^a1 +b1; a2 +b2
^L?í1￥.
10, è
a
ü/
ò
5
^p￥:
(1)?_
Fa1; a2; ￠￠￠ ; am
^L?M1￥,5a1 V?a2; ￠￠￠ ; amL?V
U.
(2)?μ??10￥
1;?2; ￠￠￠ ;?m
P
1a1 +￠￠￠+?mam +?1b1 +￠￠￠+?mbm = 0
? ?,5a1; ￠￠￠ ; amL?M1,b1; ￠￠￠ ; bmgL?M1.
(3)?oμ1;?2; ￠￠￠ ;?m?10
H,?
T
1a1 +￠￠￠+?mam +?1b1 +￠￠￠+?mbm = 0
?
? ?,5a1; ￠￠￠ ; amL?í1,b1; ￠￠￠ ; bmgL?í1.
(4)?a1; ￠￠￠ ; amL?M1,b1; ￠￠￠ ; bmgL?M1,5μ??10￥
1;?2; ￠￠￠ ;?m
P
1a1 +￠￠￠+?mam = 0;?1b1 +￠￠￠+?mbm = 0
]
H? ?.
3,(1)
!
a1 = e1 = (1; 0; 0; ￠￠￠ ; 0);
a2 = a3 = ￠￠￠ = am = 0;
@a1; a2; ￠￠￠ ; amL?M1,?a1?
?a2; ￠￠￠ ; amL?V
U.
(2)?μ??1
,￥
1;?2; ￠￠￠ ;?m
P
1a1 +￠￠￠+?mam +?1b1 +￠￠￠+?mbm = 0; (4.3)

T V1
1(a1 +b1)+￠￠￠+?m(am +bm) = 0,(4.4)
|
a1 = e1 =?b1; a2 = e2 =?b2; ￠￠￠ ; am = em =?bm;
?e1; ￠￠￠ ; em1?ê_
,5(4.4)
T? ?,7a1; ￠￠￠ ; amL?í1,b1; ￠￠￠ ; bmgL?í1.
(3)??1;?2; ￠￠￠ ;?m?10
H,
1a1 +￠￠￠+?mam +?1b1 +￠￠￠+?mbm = 0;
¤a1 +b1; a2 +b2; ￠￠￠ ; am +bmL?í1.
|a1 = a2 = ￠￠￠ = am = 0,|b1; ￠￠￠ ; bm1L?í1F,
@[
Hq.??
aa1; a2; ￠￠￠ ; am
^
L?í1￥.
(4)|a1 = (1; 0)T,a2 = (2; 0)T,b1 = (0; 3)T,b2 = (0; 4)T,
1a1 +?2a2 = 0 )?1 =?2?2
1b1 +?2b2 = 0 )?1 =?34?2
)
)?1 =?2 = 0:
D5
!
±.
11,
!b1 = a1 +a2,b2 = a2 +a3,b3 = a3 +a4,b4 = a4 +a1,￡
ü_
Fb1; b2; b3; b4L?M1.
￡
ü:
!μx1; x2; x3; x4
P¤
x1b1 +x2b2 +x3b3 +x4b4 = 0; (4.5)
52?
1c_
F￥L?M1?
5x1(a1 +a2)+x2(a2 +a3)+x3(a3 +a4)+x4(a4 +a1) = 0,'
(x1 +x4)a1 +(x1 +x2)a2 +(x2 +x3)a3 +(x3 +x4)a4 = 0,(4.6)
(a)?a1; a2; a3; a4L?M1,5(4.6)
T?x1 +x4,x1 +x2,x2 +x3,x3 +x4??1
,.
V7x1; x2; x3; x4??1
,,'b1; b2; b3; b4L?M1.
(b)?a1; a2; a3; a4L?í1,5(4.6)
T?
8>
>>>
<
>>>>
:
x1 +x4 = 0;
x1 +x2 = 0;
x2 +x3 = 0;
x3 +x4 = 0:
(4.7)
?fl
flfl
flfl
flfl
flfl
fl
1 0 0 1
1 1 0 0
0 1 1 0
0 0 1 1
flfl
flfl
flfl
flfl
flfl
= 0;
?
QZ?F(4.7)id
,3.
V7i??1
,￥
x1; x2; x3; x4
P¤(4.5)
T? ?,'b1; b2; b3; b4L?M1.
8
¤￡.

￡
üo
^B?B?
±
^.
L??5¨B ü V[￡
ü
:y1
b1 = b2?b3 +b4;
[b1; b2; b3; b4L?M1.
12,
!b1 = a1; b2 = a1 +a2; ￠￠￠ ; br = a1 +a2 +￠￠￠+ar,O_
Fa1; a2; ￠￠￠ ; arL?í1,￡
ü_
Fb1; b2; ￠￠￠ ; brL?í1.
￡
ü:
!
k1b1 +k2b2 +￠￠￠+krbr = 0; (4.8)
5
(k1 +￠￠￠+kr)a1 +(k2 +￠￠￠+kr)a2 +￠￠￠+(ki +￠￠￠+kr)ai +￠￠￠+krar = 0,(4.9)
y_
Fa1; a2; ￠￠￠ ; arL?í1,#
8>
>>><
>>>
>:
k1 +k2 +￠￠￠+kr = 0;
k2 +￠￠￠+kr = 0;
￠￠￠￠￠￠￠￠￠￠￠￠
kr = 0:
(4.10)
y1
flfl
flfl
flfl
flfl
flfl
fl
1 1 ￠￠￠ 1
0 1 ￠￠￠ 1
...,..,..
0 0 ￠￠￠ 1
flfl
flfl
flfl
flfl
flfl
fl
= 1 6= 0,#Z?F(4.10)oμ
,3,(?YVí}°¤ V¤k1 = k2 = ￠￠￠ =
kr = 0.)
',1
P(4.8)? ??O??k1 = k2 = ￠￠￠ = kr = 0.
[b1; b2; ￠￠￠ ; brL?í1.
L?}
(]6
1?)5? Is?53
6￡.?5
!?_
Fb1; b2; ￠￠￠ ; br V?_
Fa1; a2; ￠￠￠ ; arL?V
U;?a1 = b1,a2 = b2?b1,￠￠￠,
ar = br?br?1,?_
Fa1; a2; ￠￠￠ ; ar V?_
Fb1; b2; ￠￠￠ ; brL?V
U.
[_
Fa1; a2; ￠￠￠ ; ar
D_
Fb1; b2; ￠￠￠ ; br?N.V7,
R(b1; b2; ￠￠￠ ; br) = R(a1; a2; ￠￠￠ ; ar):
?a1; a2; ￠￠￠ ; arL?í1,?
R(b1; b2; ￠￠￠ ; br) = R(a1; a2; ￠￠￠ ; ar) = r:
[,b1; b2; ￠￠￠ ; brL?í1.
6￡.: ?B = (b1; b2; ￠￠￠ ; br),5
B =(b1; b2; ￠￠￠ ; br) = (a1; a1 +a2; ￠￠￠ ; a1 +a2 +￠￠￠+ar)
cr?cr?1
^￠￠￠
c2?c1
(a1; a2; ￠￠￠ ; ar);
V7,
R(b1; b2; ￠￠￠ ; br) = R(a1; a2; ￠￠￠ ; ar):
[/]
B￡E,?b1; b2; ￠￠￠ ; brL?í1.
13,p/
_
F￥?,ipB?Kví1F:
(1) a1 =
0
BB
BB
@
1
2
1
4
1
CC
CC
A
,a2 =
0
BB
BB
@
9
100
10
4
1
CC
CC
A
,a3 =
0
BB
BB
@
2
4
2
8
1
CC
CC
A;
(2) aT1 = (1; 2; 1; 3),aT2 = (4;?1;?5;?6),aT3 = (1;?3;?4;?7).
3,(1)?i2a1 = a3,
[a1,a3L?M1.
?0
BB
@
aT1
aT2
aT3
1
CC
A =
0
BB
@
1 2?1 4
9 100 10 4
2?4 2?8
1
CC
A
r3 + 2r1
^r
2?9r1
0
BB
@
1 2?1 4
0 82 19?32
0 0 0 0
1
CC
A;
?_
Fa1; a2; a3?12,BFKvL?í1F1a1; a2 (?a2; a3).
(2)?
0
BB
@
aT1
aT2
aT3
1
CC
A =
0
BB
@
1 2 1 3
4?1?5?6
1?3?4?7
1
CC
A
r3?r1
^r
2?4r1
0
BB
@
1 2 1 3
0?9?9?18
0?5?5?10
1
CC
A^
0
BB
@
1 2 1 3
0?9?9?18
0 0 0 0
1
CC
A;
?_
FaT1,aT2,aT3￥?12,KvL?í1F1aT1,aT2,
14, ?¨??MDp/
?￥
_
F￥B?Kví1F,iü?
_
¨Kví1FL?V
U:
(1)
0
BB
BB
@
25 31 17 43
75 94 53 132
75 94 54 134
25 32 20 48
1
CC
CC
A; (2)
0
BB
BB
@
1 1 2 2 1
0 2 1 5?1
2 0 3?1 3
1 1 0 4?1
1
CC
CC
A
.
54?
1c_
F￥L?M1?
3,(1)
0
BB
BB
@
25 31 17 43
75 94 53 132
75 94 54 134
25 32 20 48
1
CC
CC
A
r2?3r1
^r3?3r1
r4?r1
0
BB
BB
@
25 31 17 43
0 1 2 3
0 1 3 5
0 1 3 5
1
CC
CC
A
r4?r3
^r
3?r2
0
BB
BB
@
25 31 17 43
0 1 2 3
0 0 1 3
0 0 0 0
1
CC
CC
A;
[?1a2a3
?B?Kví1F.
(2)
0
BB
BB
@
1 1 2 2 1
0 2 1 5?1
2 0 3?1 3
1 1 0 4?1
1
CC
CC
A
r3?2r1
^r
4?r1
0
BB
BB
@
1 1 2 2 1
0 2 1 5?1
0?2?1?5 1
0 0?2 2?2
1
CC
CC
A
r3 + r2
^r
3 $ r4
0
BB
BB
@
1 1 2 2 1
0 2 1 5?1
0 0?2 2?2
0 0 0 0 0
1
CC
CC
A;
[?1a2a3
?B?Kví1F.
15,
!_
F0
BB
@
a
3
1
1
CC
A;
0
BB
@
2
b
3
1
CC
A;
0
BB
@
1
2
1
1
CC
A;
0
BB
@
2
3
1
1
CC
A
￥?12,pa,b.
3:GQ:?4?_
1a1; a2; a3; a4,i:N_
F1A.^na3; a4L?í1,7X?_
FA￥
?12,
[a3; a4
^_
FA￥B?Kví1F.5a1; a2 V[?_
Fa3; a4L?V
U.
!0
BB
@
a
3
1
1
CC
A = x
0
BB
@
1
2
1
1
CC
A+y
0
BB
@
2
3
1
1
CC
A;
¤x = 0,y = 1,V7a = 2.
¨]"￥ZE V[9
¤b = 5.
16,
!a1; a2; ￠￠￠ ; an
^BFn?_
,X?n??êUS_
e1; e2; ￠￠￠ ; en
?
ìL?V
U,￡
üa1; a2; ￠￠￠ ; anL?í1.
￡
ü:?^
!:
e1 = k11a1 +k12a2 +￠￠￠+k1nan;
e2 = k21a1 +k22a2 +￠￠￠+k2nan;
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
en = kn1a1 +kn2a2 +￠￠￠+knnan:
'0
BB
BB
B@
eT1
eT2
...
eTn
1
CC
CC
CA =
0
BB
BB
@
k11 k12 ￠￠￠ k1n
k21 k22 ￠￠￠ k2n
￠￠￠￠￠￠￠￠￠￠￠￠
kn1 kn2 ￠￠￠ knn
1
CC
CC
A
0
BB
BB
B@
aT1
aT2
...
aTn
1
CC
CC
CA:
H|?

T,¤fl
flfl
flfl
flfl
flfl
flfl
eT1
eT2
...
eTn
flfl
flfl
flfl
flfl
flfl
fl
=
flfl
flfl
flfl
flfl
flfl
k11 k12 ￠￠￠ k1n
k21 k22 ￠￠￠ k2n
￠￠￠￠￠￠￠￠￠￠￠￠
kn1 kn2 ￠￠￠ knn
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
fl
aT1
aT2
...
aTn
flfl
flfl
flfl
flfl
flfl
fl
:
L?}
(]6
1?)5? Is?55
?flfl
flfl
flfl
flfl
flfl
fl
eT1
eT2
...
eTn
flfl
flfl
flfl
flfl
flfl
fl
= 1 6= 0; =)
flfl
flfl
flfl
flfl
flfl
fl
aT1
aT2
...
aTn
flfl
flfl
flfl
flfl
flfl
fl
6= 0;
'n?_
Fa1; a2; ￠￠￠ ; an
? ?￥?1n.
[a1; a2; ￠￠￠ ; anL?í1.
6￡._
Fe1; e2; ￠￠￠ ; en
?_
Fa1; a2; ￠￠￠ ; anL?V
U,5
R(e1; e2; ￠￠￠ ; en) 6 R(a1; a2; ￠￠￠ ; an),(P.86? ?3)
7
R(e1; e2; ￠￠￠ ; en) = n;OR(a1; a2; ￠￠￠ ; an) 6 n;
[
R(a1; a2; ￠￠￠ ; an) = n:
¤a1; a2; ￠￠￠ ; anL?í1.
17,
!a1; a2; ￠￠￠ ; an
^BFn?_
,￡
ü
ìL?í1￥ sA1Hq
^:?Bn?_
? V?
ìL?V
U.
￡
ü,( s?)
!?Bn?_
? V?a1; a2; ￠￠￠ ; anL?V
U,5n??ê_
Fe1; e2; ￠￠￠ ; en
?_
Fa1; a2; ￠￠￠ ; anL?V
U.?
B5¤a1; a2; ￠￠￠ ; anL?í1.
(A1?)
!a1; a2; ￠￠￠ ; anL?í1.?ón?_
b,5_
Fa1; a2; ￠￠￠ ; an; bL?M1(n+1?
n?_

^L?M1￥).
?P.90? ?5(3),5_
bA
?_
Fa1; a2; ￠￠￠ ; anL?V
U(OV
U
T
^?B￥).
18,
!_
Fa1; a2; ￠￠￠ ; amL?M1,Oa1 6= 0,￡
üi
?_
ak (2 6 k 6 m),
Pak
?
a1; ￠￠￠ ; ak?1L?V
U.
￡
ü:L
!?i?"￥ak.5a2?
?a1L?V
U,V7_
Fa1; a2L?í1.
a3?
?a1; a2L?V
U,?_
Fa1; a2L?í1,
[_
Fa1,a2,a3L?í1.
GQ ?w, V[¤?_
Fa1; a2; ￠￠￠ ; amL?í1.?D5
!
±.L
!?? ?.¤￡.
19,
!_
FB,b1; ￠￠￠ ; br
?_
FA,a1; ￠￠￠ ; asL?V
U1
(b1; ￠￠￠ ; br) = (a1; ￠￠￠ ; as)K;
?K1s￡r ?,OAFL?í1.￡
üBFL?í1￥ sA1Hq
^ ?K￥?R(K) = r.
￡
ü,(A1?)
!BFL?í1.
:B = (b1; ￠￠￠ ; br),A = (a1; ￠￠￠ ; as)5μ
B = AK,(4.11)
??￥?é?
R(B) = R(AK) 6 R(K),(4.12)
7?BFL?í1?R(B) = r,#R(K) > r.
?K1r￡s¨ ?,5R(K) 6 minfr; sg6 r.
8
?R(K) = r.
( s?)?R(K) = r.
7
x1b1 +x2b2 +￠￠￠+xrbr = 0,(4.13)
56?
1c_
F￥L?M1?
/￡Z?(4.13)oμ
,3.1ZL:Z?(4.13)1
Bx = 0,(4.14)
}?(4.11)
T5μ
AKx = 0,(4.15)
?_
FA,a1; ￠￠￠ ; asL?í1,μR(A) = r.
[Z?(4.15)oμ
,3:
Kx = 0,(4.16)
?R(K) = r,
[Z?(4.16)oμ
,3:
x = 0:
[b1; b2; ￠￠￠ ; brL?í1.
20,
!8
>>>>
<
>>>>
:
fl1 = fi2 +fi3 +￠￠￠+fin;
fl2 = fi1 +fi3 +￠￠￠+fin;
￠￠￠￠￠￠
fln = fi1 +fi2 +￠￠￠+fin?1:
￡
ü_
Ffi1; fi2; ￠￠￠ ; finD_
Ffl1; fl2; ￠￠￠ ; fln?N.
￡
ü:?5
!?_
Ffl1; fl2; ￠￠￠ ; fln V[?_
Ffi1; fi2; ￠￠￠ ; finL?V
U./
o3￡
ü_
F
fi1; fi2; ￠￠￠ ; fin V?_
Ffl1; fl2; ￠￠￠ ; flnL?V
U' V.
?5
!¤
fl1 +fl2 +￠￠￠+fln = (n?1)(fi1 +fi2 +￠￠￠+fin);
[
(fl1 +fl2 +￠￠￠+fln)=(n?1) = fi1 +fi2 +￠￠￠+fin:
¤8
>>>>
<
>>>>
:
fi1 = (fl1 +fl2 +￠￠￠+fln)=(n?1)?fl1;
fi2 = (fl1 +fl2 +￠￠￠+fln)=(n?1)?fl2;
￠￠￠￠￠￠
fin = (fl1 +fl2 +￠￠￠+fln)=(n?1)?fln:
¤￡_
Ffi1; fi2; ￠￠￠ ; fin V?_
Ffl1; fl2; ￠￠￠ ; flnL?V
U.
8
,_
Ffi1; fi2; ￠￠￠ ; finD_
Ffl1; fl2; ￠￠￠ ; fln?N.
21,X?3¨ ?AD3?
_
x
@A3x = 3Ax?A2x,O_
Fx,Ax,A2xL?í1,
(1):P = (x; Ax; A2x),p3¨ ?B,
PAP = PB; (2)pjAj.
3,(1)?P = (x; Ax; A2x),μ
AP = A(x; Ax; A2x)
= (Ax; A2x; A3x)
= (x; Ax; A2x)
0
BB
@
0 0 0
1 0 3
0 1?2
1
CC
A (?A3x = 3Ax?A2x)
= P
0
BB
@
0 0 0
1 0 3
0 1?2
1
CC
A:
L?}
(]6
1?)5? Is?57
?i? ?P
^3¨Z?,?_
Fx,Ax,A2xL?í1,
[ ?P V
I.¤
B = P?1AP = P?1P
0
BB
@
0 0 0
1 0 3
0 1?2
1
CC
A =
0
BB
@
0 0 0
1 0 3
0 1?2
1
CC
A:
(2)?A = PBP?1,
H|?

T¤,
jAj = jBj = 0:
22,p/

QL?Z?F￥$3":
(1)
8
>><
>>:
x1?8x2 +10x3 +2x4 = 0;
2x1 +4x2 +5x3?x4 = 0;
3x1 +8x2 +6x3?2x4 = 0:
(2)
8
>><
>>:
2x1?3x2?2x3 +x4 = 0;
3x1 +5x2 +4x3?2x4 = 0;
8x1 +7x2 +6x3?3x4 = 0:
(3)nx1 +(n?1)x2 +￠￠￠2xn?1 +xn = 0.
3,(1)
A =
0
BB
@
1?8 10 2
2 4 5?1
3 8 6?2
1
CC
A
r2?2r1
^r
3?3r1
0
BB
@
1?8 10 2
0 20?15?5
0 32?24?8
1
CC
A
r2 ￥4
^r
3 ￥8
0
BB
@
1?8 10 2
0 4?3?1
0 4?3?1
1
CC
A
r3?r2
^r
1 + 2r2
0
BB
@
1 0 4 0
0 4?3?1
0 0 0 0
1
CC
A
r2 ￥4
^
0
BB
@
1 0 4 0
0 1?34?14
0 0 0 0
1
CC
A:
[eZ?F?N?
(
x1 =?4x3;
x2 = 34x3 + 14x4;'
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
= c1
0
BB
BB
@
4
3
4
1
0
1
CC
CC
A
+c2
0
BB
BB
@
0
1
4
0
1
1
CC
CC
A
:
yN$3"1?1 =
0
BB
BB
@
4
3
4
1
0
1
CC
CC
A;?2 =
0
BB
BB
@
0
1
4
0
1
1
CC
CC
A
,?1?1 =
0
BB
BB
@
16
3
4
0
1
CC
CC
A;?2 =
0
BB
BB
@
0
1
0
4
1
CC
CC
A
.
(2)
A =
0
BB
@
2?3?2 1
3 5 4?2
8 7 6?3
1
CC
A
r3?r1?2r2
^2r
2?3r1
0
BB
@
2?3?2 1
0 19 14?7
0 0 0 0
1
CC
A
r1 ￥2 + 338r2
^r
2 ￥19
0
BB
@
1 0 219? 119
0 1 1419? 719
0 0 0 0
1
CC
A:
[eZ?F?N?
(
x1 =? 219x3 + 119x4;
x2 =?1419x3 + 719x4:'
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
= c1
0
BB
BB
@
219
1419
1
0
1
CC
CC
A
+c2
0
BB
BB
@
1
19
7
19
0
1
1
CC
CC
A
:
58?
1c_
F￥L?M1?
yN$3"1?1 =
0
BB
BB
@
219
1419
1
0
1
CC
CC
A;?2 =
0
BB
BB
@
1
19
7
19
0
1
1
CC
CC
A
,?1?1 =
0
BB
BB
@
2
14
19
0
1
CC
CC
A;?2 =
0
BB
BB
@
1
7
0
19
1
CC
CC
A
.
(3)eZ?'1
xn =?nx1?(n?1)x2?￠￠￠?2xn?1:
?8
>>>>
>>>
<
>>>>
>>>:
x1 = x1;
x2 = x2;
...
xn?1 = xn?1;
xn =?nx1?(n?1)x2?￠￠￠?2xn?1:
[$3"1
(?1;?2; ￠￠￠ ;?n?1) =
0
BB
BB
BB
B@
1 0 ￠￠￠ 0
0 1 ￠￠￠ 0
...,..,..
0 0 ￠￠￠ 1
n?n+1 ￠￠￠?2
1
CC
CC
CC
CA
:
23,
!A =
2?2 1 3
9?5 2 8
!
,pB?4￡2 ?B,
PAB = O,O
R(B) = 2:
3:??R(B) = 2,
[ V
!B =
0
BB
BB
@
1 0
0 1
x1 x2
x3 x4
1
CC
CC
A
.5?
AB =
2?2 1 3
9?5 2 8
!
0
BB
BB
@
1 0
0 1
x1 x2
x3 x4
1
CC
CC
A
=
0 0
0 0
!;
V¤0
BB
BB
@
1 0 3 0
0 1 0 3
2 0 8 0
0 2 0 8
1
CC
CC
A
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
=
0
BB
BB
@
2
2
9
5
1
CC
CC
A;
3Nd
QL?Z?F V¤?B3
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
=
0
BB
BB
@
11
2
1
2
52
1
2
1
CC
CC
A;
L?}
(]6
1?)5? Is?59
#
p ?
B =
0
BB
BB
@
1 0
0 1
11
2
1
2
52 12
1
CC
CC
A;?
0
BB
BB
@
2 0
0 2
11 1
5 1
1
CC
CC
A
:
N5?
@Hq￥ ?BA???B?.1?
AB = O
H]
H·e
?? ?,5?
T·H￥O?M,7 ?B$é?
MD7?
3
?M.
?9
^1
I
1üB
!10
BB
BB
@
1 0
0 1
x1 x2
x3 x4
1
CC
CC
A;
7?31
!10
BB
BB
@
x1 x2
x3 x4
x5 x6
x7 x8
1
CC
CC
A
￥ey.
24,pB?
QL?Z?F,
P
￥$3"1
1 = (0; 1; 2; 3)T;?2 = (3; 2; 1; 0)T:
3:A?eZ?F￥Y31
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
= k1
0
BB
BB
@
0
1
2
3
1
CC
CC
A
+k2
0
BB
BB
@
3
2
1
0
1
CC
CC
A; (k1; k2 2R):
'8
>>>>
<
>>>
>:
x1 = 3k2;
x2 = k1 +2k2;
x3 = 2k1 +k2;
x4 = 3k1:
h?k1; k2¤(
2x1?3x2 +x4 = 0;
x1?3x3 +2x4 = 0:
N'
p￥
QL?Z?F.
25,
!
1í
QL?Z?F
I,
(
x1 +x2 = 0;
x2?x4 = 0; II,
(
x1?x2 +x3 = 0;
x2?x3 +x4 = 0:
p,(1)Z?FIDII￥$3"; (2) IDII￥
3.
60?
1c_
F￥L?M1?
3,(1)y1
I ()
(
x1 =?x2;
x4 = x2; ()
8
>>>>
<
>>>
>:
x1 =?x2;
x2 = x2;
x3 = x3;
x4 = x2;
[I￥$3"1
1 =
0
BB
BB
@
1
1
0
1
1
CC
CC
A;?2 =
0
BB
BB
@
0
0
1
0
1
CC
CC
A
:
?
II ()
(
x1 = x2?x3;
x4 =?x2 +x3; ()
8>
>>><
>>>
>:
x1 = x2?x3;
x2 = x2;
x3 = x3;
x4 =?x2 +x3;
[II￥$3"1
1 =
0
BB
BB
@
1
1
0
1
1
CC
CC
A;?2 =
0
BB
BB
@
1
0
1
1
1
CC
CC
A
:
(2) ó ?Z?FIII¤8
>>>>
<
>>>
>:
x1 +x2 = 0;
x2?x4 = 0;
x1?x2 +x3 = 0;
x2?x3 +x4 = 0:
?0
BB
BB
@
1 1 0 0
0 1 0?1
1?1 1 0
0 1?1 1
1
CC
CC
A
r3?r1
^r
4 + r2
0
BB
BB
@
1 1 0 0
0 1 0?1
0?2 1 0
0 2?1 0
1
CC
CC
A
r4 + r3
^
0
BB
BB
@
1 1 0 0
0 1 0?1
0 2?1 0
0 0 0 0
1
CC
CC
A;
'8
>>>
><
>>>
>:
x1 =?x2;
x2 = x2;
x3 = 2x2;
x4 = x2;
¤Z?FIDII￥
31
x = c
0
BB
BB
@
1
1
2
1
1
CC
CC
A; (c 2R):
26,
!n¨ ?A
@A2 = A,E1n¨?ê ?,￡
R(A)+R(A?E) = n:
L?}
(]6
1?)5? Is?61
4
U: ?¨ ??é68.
￡
ü:y1A(A?E) = A2?A = A?A = O,
[? ??￥?é8(P.71),?
R(A)+R(A?E) 6 n:
?? ??￥?é6(P.71),μ
R(A)+R(A?E) = R(A)+R(E?A) > R(A+E?A) = R(E) = n:
[R(A)+R(A?E) = n.
27,
!A1n¨ ?(n > 2),A?1A￥?
?,￡
ü
R(A?) =
8>
>><
>>>:
n;?R(A) = n;
1;?R(A) = n?1;
0;?R(A) 6 n?2:
3,(1)?R(A) = n.?A1n¨Z?,? ?A V
I.V7 ?A? V
I,¤R(A?) = n.
(2)?R(A) = n?1.5 ?Aà
iB?n?1¨d
,0
T,V7 ?A??à
μB?í
íd
,,
¤
R(A?) > 1,(4.17)
??R(A) = n?1?jAj = 0,
[
AA? = jAjE = O:
? ??é8?
R(A)+R(A?) 6 n:
}?R(A) = n?1,¤
R(A?) 6 1,(4.18)
8(4.17)
T(4.18)
T¤
R(A?) = 1:
(3)?R(A) 6 n?2.5 ?A￥
[n?1¨0
T?1
,,?
P¤A? = O.
[
R(A?) = 0:
28,p/
d
QZ?F￥B?3#?￥
QL?Z?F￥$3":
(1)
8
>><
>>:
x1 + x2 = 5;
2x1 + x2 + x3 +2x4 = 1;
5x1 +3x2 +2x3 +2x4 = 3;
(2)
8
>><
>>:
x1?5x2 +2x3?3x4 = 11;
5x1 +3x2 +6x3?x4 =?1;
2x1 +4x2 +2x3 +x4 =?6:
3,(1)
B =
0
BB
@
1 1 0 0 5
2 1 1 2 1
5 3 2 2 3
1
CC
A
r2?2r1
^r
3?5r1
0
BB
@
1 1 0 0 5
0?1 1 2?9
0?2 2 2?22
1
CC
A
r3?2r2
^r
2 ￡(?1)
0
BB
@
1 1 0 0 5
0 1?1?2 9
0 0 0?2?4
1
CC
A
r2?r3
^r
3 ￥?2
0
BB
@
1 1 0 0 5
0 1?1 0 13
0 0 0 1 2
1
CC
A
r1?r2
^
0
BB
@
1 0 1 0?8
0 1?1 0 13
0 0 0 1 2
1
CC
A:
62?
1c_
F￥L?M1?
[
· =
0
BB
BB
@
8
13
0
2
1
CC
CC
A;? =
0
BB
BB
@
1
1
1
0
1
CC
CC
A
:
(2)
B =
0
BB
@
1?5 2?3 11
5 3 6?1?1
2 4 2 1?6
1
CC
A
r2?r1?2r3
^r
3?2r1
0
BB
@
1?5 2?3 11
0 0 0 0 0
0 14?4 7?28
1
CC
A
r3 ￥14
^r
3 $ r2
0
BB
@
1?5 2?3 11
0 1?17 12?2
0 0 0 0 0
1
CC
A
r1 + 5r2
^
0
BB
@
1 0 97?12 1
0 1?17 12?2
0 0 0 0 0
1
CC
A:
[
· =
0
BB
BB
@
1
2
0
0
1
CC
CC
A;?1 =
0
BB
BB
@
9
1
7
0
1
CC
CC
A;?2 =
0
BB
BB
@
1
1
0
2
1
CC
CC
A
:
29,
!
1íd
QL?Z?F￥"
?￥?13,X?·1,·2,·3
^
￥??3_
.O
·1 =
0
BB
BB
@
2
3
4
5
1
CC
CC
A; ·2 +·3 =
0
BB
BB
@
1
2
3
4
1
CC
CC
A
p?Z?F￥Y3.
3:?? ?￥?13,Z?Fμ4???
,n?r = 4?3 = 1,#?￥
QL?Z?F￥$3
"cμB?_
,O??·1,·2,·3 (1Z?F￥3,?d
QL?Z?F3￥2?é¤
2·1?(·2 +·3) = (·1?·2)+(·1?·3) =
0
BB
BB
@
3
4
5
6
1
CC
CC
A
:
1$3"_
,?·1?·2,·1?·31?￥
QZ?F￥3.#NZ?F￥Y31:
x = c
0
BB
BB
@
3
4
5
6
1
CC
CC
A
+
0
BB
BB
@
2
3
4
5
1
CC
CC
A; (c 2R):
30,
!μ_
FA,a1 =
0
BB
@
fi
2
10
1
CC
A; a2 =
0
BB
@
2
1
5
1
CC
A; a3 =
0
BB
@
1
1
4
1
CC
A,#_
b =
0
BB
@
1
fl
1
1
CC
A,ùfi,fl1
′
H
(1)_
b?
?_
FAL?V
U;
(2)_
b
μ?
FAL?V
U,OV
U
T?B;
L?}
(]6
1?)5? Is?63
(3)_
b
μ?
FAL?V
U,OV
U
T??B,ipB?V
U
T.
3:
!
x1a1 +x2a2 +x3a3 = b;
'8
>><
>>:
fix1?2x2?x3 = 1;
2x1 +x2 +x3 = fl;
10x1 +5x2 +4x3 =?1:
(4.19)
a/)
Z?F(4.19)￥3' V.y1
(A; b) =
0
BB
@
fi?2?1 1
2 1 1 fl
10 5 4?1
1
CC
A
r1 + r2
^r
3?4r2
0
BB
@
fi+2?1 0 1+fl
2 1 1 fl
2 1 0?1?4fl
1
CC
A
r1 + r3
^r
2?r3
0
BB
@
fi+4 0 0?3fl
0 0 1 5fl +1
2 1 0?1?4fl
1
CC
A
(1)?fi + 4 = 0O?3fl 6= 0
H,R(A) = 2 6= R(A; b) = 3,Z?F(4.19)í3.',?fi =?4Ofl 6= 0
H,_
b?
?_
FAL?V
U;
(2)?fi + 4 6= 0
H,R(A) = R(A; b) = 3,Z?F(4.19)μ?B3.',?fi 6=?4
H,_
b
μ?

FAL?V
U,OV
U
T?B;
(3)?fi+4 = 0O?3fl = 0
H,R(A) = R(A; b) = 2 6 3,Z?F(4.19)μíK3.N
HZ?F(4.19)
?N?
(
2x1 +x2 =?1;
x3 = 1,)
(
x2 =?2x1?1;
x3 = 1,)
8>
><
>>:
x1 = c;
x2 =?2c?1;
x3 = 1:
',?fi =?4Ofl = 0
H,_
b
μ?
FAL?V
U,OV
U
T??B,B?V
U
T1
b = ca1?(2c+1)a2 +a3; (c 2R):
31,
!
a =
0
BB
@
a1
a2
a3
1
CC
A; b =
0
BB
@
b1
b2
b3
1
CC
A; c =
0
BB
@
c1
c2
c3
1
CC
A;
￡
ü?°L8
>><
>>:
l1,a1x+b1y +c1 = 0;
l2,a2x+b2y +c2 = 0; (a2i +b2i 6= 0; i = 1; 2; 3)
l3,a3x+b3y +c3 = 0:
M??B?￥ sA1Hq1:_
Fa; bL?í1,O_
Fa; b; cL?M1.
￡
ü:?°LM??B?￥ sA1Hq1Z?F
8>
><
>>:
a1x+b1y +c1 = 0;
a2x+b2y +c2 = 0; (a2i +b2i 6= 0; i = 1; 2; 3)
a3x+b3y +c3 = 0:
(4.20)
μ?B3.:Z?F(4.20)1
xa+yb =?c,(4.21)
64?
1c_
F￥L?M1?
Z?F(4.21)μ?B3￥ 1Hq
^
R(a; b) = R(a; b; c) = 2:
'_
Fa; bL?í1,O_
Fa; b; cL?M1.
32,
! ?A = (a1; a2; a3; a4),?a2; a3; a4L?í1,a1 = 2a2?a3._
b = a1+a2+a3+a4,
pZ?Ax = b￥Y3.
3::x = (x1; x2; x3; x4)T,5Ax = b1
a1x1 +a2x2 +a3x3 +a4x4 = b:
}?b = a1 +a2 +a3 +a4,a1 = 2a2?a3,? ?¤
(2x1 +x2?3)a2 +(?x1 +x3)a3 +(x4?1)a4 = 0:
?a2; a3; a4L?í1,¤8
>><
>>:
2x1 +x2?3 = 0;
x1 +x3 = 0;
x4?1 = 0:
(4.22)
Z?F(4.22)?N?
8
>><
>>:
x2 =?2x1 +3;
x3 = x1;
x4 = 1:
,
8
>>>>
<
>>>
>:
x1 = x1;
x2 =?2x1 +3;
x3 = x1;
x4 = 1:
¤Z?Ax = b￥Y31
x =
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
= c
0
BB
BB
@
1
2
1
0
1
CC
CC
A
+
0
BB
BB
@
0
3
0
1
1
CC
CC
A; (c 2R):
33,
!·?
^d
QL?Z?FAx = b￥B?3,?1; ￠￠￠ ;?n?r
^?￥
QL?Z?F￥B?$
3",￡
ü:
(1) ·?;?1; ￠￠￠ ;?n?rL?í1;
(2) ·?; ·? +?1; ￠￠￠ ; ·? +?n?rL?í1.
￡
ü,(1)L
!·?;?1; ￠￠￠ ;?n?rL?M1.7?$3"￥?l,1; ￠￠￠ ;?n?r
^L?í1￥,5·?
V[1; ￠￠￠ ;?n?rL?V
U.V7·?
^
QZ?Ax = 0￥3,?D·?
^d
QL?Z?FAx = b
￥3
±.
[L
!?? ?.'·?;?1; ￠￠￠ ;?n?rL?í1.
(2)^?_
F·?;?1; ￠￠￠ ;?n?rD_
F·?; ·? +?1; ￠￠￠ ; ·? +?n?r?N.??'5(1)￥2
,
·?;?1; ￠￠￠ ;?n?rL?í1,?
R(·?; ·? +?1; ￠￠￠ ; ·? +?n?r) = R(·?;?1; ￠￠￠ ;?n?r) = n?r +1:
[,·?; ·? +?1; ￠￠￠ ; ·? +?n?rL?í1.
6￡,(1)Q￡E.L
!·?;?1; ￠￠￠ ;?n?rL?M1,5i"??10￥
k0; k1; ￠￠￠ ; kn?r
P¤
k0·? +k1?1 +￠￠￠+kn?r?n?r = 0,(4.23)
?,k0 6= 0,?5,?1; ￠￠￠ ;?n?rL?M1,?D$3"
^L?í1￥á
3
±.
L?}
(]6
1?)5? Is?65
??·?1+3,?1; ￠￠￠ ;?n?r1$3",#¤
A(k0·? +k1?1 +￠￠￠+kn?r?n?r) = k0A·? = k0b:
7?(4.23)
T V¤
A(k0·? +k1?1 +￠￠￠+kn?r?n?r) = 0:
#b = 0,75?Z?F1d
QL?Z?F,μb 6= 0.
D5
!á
3
±,L
!?? ?,#·?;?1; ￠￠￠ ;?n?rL?í1.
(2)Q￡E.L
!·?; ·? +?1; ￠￠￠ ; ·? +?n?rL?M1.5i"??1
,￥
k0; k1; ￠￠￠ ; kn?r
P¤
k0·? +k1(·? +?1)+￠￠￠+kn?r(·? +?n?r) = 0,(4.24)
'(k0 +k1 +￠￠￠+kn?r)·? +k1?1 +￠￠￠+kn?r?n?r = 0.
?k0+k1+￠￠￠+kn?r = 0,?1; ￠￠￠ ;?n?r
^L?í1￥BF$3",#k0 = k1 = ￠￠￠ = kn?r = 0,
?(4.24)
T¤k0 = 0,N
H
k0 = k1 = ￠￠￠ = kn?r = 0;
DL
!
±.
?k0 +k1 +￠￠￠+kn?r 6= 0?'5(1)?,·?;?1; ￠￠￠ ;?n?rL?í1,#
k0 +k1 +￠￠￠+kn?r = k1 = k2 = ￠￠￠ = kn?r = 0;
DL
!
±,
8
,L
!?? ?,e
5¤￡.
34,
!·1; ￠￠￠ ; ·s
^d
QL?Z?FAx = b￥s?3,k1; ￠￠￠ ; ks1
L
,
@k1+k2+￠￠￠+ks = 1.
￡
ü
x = k1·1 +k2·2 +￠￠￠+ks·s
9
^
￥3.
￡
ü:??·1; ￠￠￠ ; ·s
^d
QL?Z?FAx = b￥s?3.#μA·i = b,(i = 1; ￠￠￠ ; s).
7
A(k1·1 +k2·2 +￠￠￠+ks·s)
=k1A·1 +k2A·2 +￠￠￠+ksA·s
=b(k1 +￠￠￠+ks) = b;
'Ax = b,(x = k1·1 +k2·2 +￠￠￠+ks·s).
V7x = k1·1 +k2·2 +￠￠￠+ks·s9
^Z?￥3.
35,
!d
QL?Z?FAx = b￥"
?￥?1r,·1; ￠￠￠ ; ·n?r+1
^
￥n?r + 1?L?í1
￥3(?533?
?μn?r +1?L?í1￥3).
k￡
￥?B3 VV
U1
x = k1·1 +k2·2 +￠￠￠+kn?r+1·n?r+1; (?k1 +￠￠￠+kn?r+1 = 1):
￡
ü:
!x1Ax = b￥?B3.X?·1; ·2; ￠￠￠ ; ·n?r+1L?í1O (1Ax = b￥3.|
1 = ·2?·1;?2 = ·3?·1; ￠￠￠ ;?n?r = ·n?r+1?·1,(4.25)
5
ì (1
QZ?Ax = 0￥3./¨Q￡E￡
ü,?1;?2; ￠￠￠ ;?n?rL?í1.
L
!?1;?2; ￠￠￠ ;?n?rL?M1,5i??1
,￥
l1; l2; ￠￠￠ ; ln?r
P¤
l1?1 +l2?2 +￠￠￠+ln?r?n?r = 0:
66?
1c_
F￥L?M1?
}?(4.25)
T? ?¤
(l1 +l2 +￠￠￠+ln?r)·1 +l1·2 +l2·3 +￠￠￠+ln?r·n?r+1 = 0:
?·1;·2;￠￠￠ ;·n?r+1L?í1,?
(l1 +l2 +￠￠￠+ln?r) = l1 = l2 = ￠￠￠ = ln?r = 0;
DL
!
±,#L
!?? ?.
[?1;?2; ￠￠￠ ;?n?rL?í1,1
QZ?Ax = 0￥BF.
??x; ·1 (1Ax = b￥3,
[x?·11
QZ?Ax = 0￥3.5x?·1 V?
1;?2; ￠￠￠ ;?n?r
L?V
U,
!
x?·1 = k2?1 +k3?2 +￠￠￠+kn?r?1?n?r
= k2(·2?·1)+k3(·3?·1)+￠￠￠+kn?r+1(·n?r+1?·1);
? ?¤
x = ·1(1?k2?k3?￠￠￠?kn?r+1)+k2·2 +k3·3 +￠￠￠+kn?r+1·n?r+1 = 0:
7k1 = 1?k2?k3?￠￠￠?kn?r+1,5k1 +k2 +k3 +￠￠￠+kn?r+1 = 1,7O
x = k1·1 +k2·2 +￠￠￠+kn?r+1·n?r+1:
￡8.
36,
!
V1 = 'x = (x1; x2; ￠￠￠ ; xn)T flfl x1; ￠￠￠ ; xn 2R
@x1 +x2 +￠￠￠+xn = 0“;
V2 = 'x = (x1; x2; ￠￠￠ ; xn)T flfl x1; ￠￠￠ ; xn 2R
@x1 +x2 +￠￠￠+xn = 1“:
ùV1,V2
^?
^_
bW?1
I
1?
￡
ü:"V?1_
bWo3
@Hq:
?fi 2 V,fl 2 V,5fi+fl 2 V;
?fi 2 V,? 2R,5?fi 2 V.
(1)?i￥fi 2 V,fl 2 V,
!
fi = (fi1; fi2; ￠￠￠ ; fin)T; fi1 +fi2 +￠￠￠+fin = 0;
fl = (fl1; fl2; ￠￠￠ ; fln)T; fl1 +fl2 +￠￠￠+fln = 0:
5fi+fl = (fi1 +fl1; fi2 +fl2; ￠￠￠ ; fin +fln)T,O
(fi1 +fl1)+(fi2 +fl2)+￠￠￠+(fin +fln)
=(fl1 +fl2 +￠￠￠+fln)+(fi1 +fi2 +￠￠￠+fin)
=0:
#
fi+fl 2 V1,(4.26)
?i￥? 2R,
fi = (?fi1;?fi2; ￠￠￠ ;?fin):
L?}
(]6
1?)5? Is?67
y1
fi1 +?fi2 +￠￠￠+?fin =?(fi1 +fi2 +￠￠￠+fin) =?￠0 = 0:
#
fi 2 V1,(4.27)
8(4.26)(4.27)
T,¤￡V1
^_
bW.
(2) V2?
^_
bW,y1:
(fi1 +fl1)+(fi2 +fl2)+￠￠￠+(fin +fln)
=(fl1 +fl2 +￠￠￠+fln)+(fi1 +fi2 +￠￠￠+fin)
=1+1
=2:
#
fi+fl =2 V2:
37,
k￡:?a1 = (0; 1; 1)T,a2 = (1; 0; 1)T,a3 = (1; 1; 0)T
3?￥_
bWü
^R3.
￡
ü:
!A = (a1; a2; a3),y1
jAj = ja1; a2; a3j =
flfl
flfl
flfl
flfl
0 1 1
1 0 1
1 1 0
flfl
flfl
flfl
flfl
=?2 6= 0:
?R(A) = 3,#a1; a2; a3L?í1.
??a1; a2; a3 (1??,O?13,
[a1; a2; a31N?? bW￥BF,#?a1; a2; a3
3?￥
_
bWü
^R3.
38,?a1 = (1; 1; 0; 0)T,a2 = (1;0; 1; 1)T
3?￥_
bW:TL1,?b1 = (2;?1; 3; 3)T,b2 =
(0; 1;?1;?1)T
3?￥_
bW:TL2,
k￡L1 = L2.
￡
ü:?^?C_
Fa1; a2D_
Fb1; b2?N.y1
a1 = 12(b1 +3b2); a2 = 12(b1 +b2);
b1 =?a1 +3a2; b2 = a1?a2:
????P.105 è23?\?N￥_
F
3?￥_
bWM]",
[L1 = L2.
39,￡a1 = (1;?1; 0)T,a2 = (2; 1; 3)T,a3 = (3; 1; 2)T1R3￥B?,iüv1 = (5; 0; 7)T,
v2 = (?9;?8;?13)T¨??L?V
U.
3:: ?A = (a1; a2; a3).??
jAj =
flfl
flfl
flfl
flfl
1 2 3
1 1 1
0 3 2
flfl
flfl
flfl
flfl
=?6 6= 0;
' ?A￥?13,#a1; a2; a3L?í1,1R3￥B?.
!
v1 = k1a1 +k2a2 +k3a3;
v2 =?1a1 +?2a2 +?3a3:
68?
1c_
F￥L?M1?
1p¤k1,k2,k3?1,?2,?3,'1pL?Z?FAx = v1Ax = v2￥3.?
(A; v1; v2) =
0
BB
@
1 2 3 5?9
1 1 1 0?8
0 3 2 7?13
1
CC
A
r2 + r1
^
0
BB
@
1 2 3 5?9
0 3 4 5?17
0 3 2 7?13
1
CC
A
r3?r2
^
0
BB
@
1 2 3 5?9
0 3 4 5?17
0 0?2 2 4
1
CC
A
r3 ￥(?2)
^
0
BB
@
1 2 3 5?9
0 3 4 5?17
0 0 1?1?2
1
CC
A
r2?4r3
^r
1?3r3
0
BB
@
1 2 0 8?3
0 3 0 9?9
0 0 1?1?2
1
CC
A
r2 ￥3
^
0
BB
@
1 2 0 8?3
0 1 0 3?3
0 0 1?1?2
1
CC
A
r1?2r2
^
0
BB
@
1 0 0 2 3
0 1 0 3?3
0 0 1?1?2
1
CC
A:
p¤Z?FAx = v1Ax = v2￥3,'
8
>><
>>:
k1 = 2;
k2 = 3;
k3 =?1;
8
>><
>>:
1 = 3;
2 =?3;
3 =?2:
#
v1 = 2a1 +3a2?a3; v2 = 3a1?3a2?2a3:
40,X?R3￥
?1
a1 =
0
BB
@
1
1
1
1
CC
A; a2 =
0
BB
@
1
0
1
1
CC
A; a3 =
0
BB
@
1
0
1
1
CC
A#b1 =
0
BB
@
1
2
1
1
CC
A; b2 =
0
BB
@
2
3
4
1
CC
A; b3 =
0
BB
@
3
4
3
1
CC
A:
p?a1; a2; a3?b1; b2; b3￥V ?.
3:?V ?￥?l?,Va1; a2; a3?b1; b2; b3￥V ?1
P = A?1B;
? úA = (a1; a2; a3),B = (b1; b2; b3).y1
(A; B) =
0
BB
@
1 1 1 1 2 3
1 0 0 2 3 4
1?1 1 1 4 5
1
CC
A
r1?r2
^r
3?r2
0
BB
@
0 1 1?1?1?1
1 0 0 2 3 4
0?1 1?1 1 1
1
CC
A
r3 + r1
^r
1 $ r2
0
BB
@
1 0 0 2 3 4
0 1 1?1?1?1
0 0 2?2 0 0
1
CC
A
r2?r3=2
^r
3 ￥2
0
BB
@
1 0 0 2 3 4
0 1 0 0?1?1
0 0 1?1 0 0
1
CC
A;
[
P = A?1B =
0
BB
@
2 3 4
0?1?1
1 0 0
1
CC
A:
??cM
?#=Q?
1,
k¨
@
á+Eü/
_
F??:
(1) (a1; a2; a3) =
0
BB
@
1 1 1
1 2 4
1 3 9
1
CC
A;
(2) (a1; a2; a3) =
0
BB
BB
@
1 1?1
0?1 1
1 0 1
1 1 0
1
CC
CC
A
3,(1)?
@
á+??ZE,¤
b1 = a1 =
0
BB
@
1
1
1
1
CC
A;
b2 = a2? [b1; a2][b
1; b1]
b1 =
0
BB
@
1
0
1
1
CC
A;
b3 = a3? [b1; a3][b
1; b1]
b1? [b2; a3][b
2; b2]
b2 = 13
0
BB
@
1
2
1
1
CC
A:
#??a¤:
(b1; b2; b3) =
0
BB
@
1?1 13
1 0?23
1 1 13
1
CC
A:
(2)?
@
á+??ZE¤
b1 = a1 =
0
BB
BB
@
1
0
1
1
1
CC
CC
A;
b2 = a2? [b1; a2][b
1; b1]
b1 = 13
0
BB
BB
@
1
3
2
1
1
CC
CC
A;
b3 = a3? [b1; a3][b
1; b1]
b1? [b2; a3][b
2; b2]
b2 = 15
0
BB
BB
@
1
3
3
4
1
CC
CC
A
:
#??a¤
(b1; b2; b3) =
0
BB
BB
@
1 13?15
0?1 35
1 23 35
1 13 45
1
CC
CC
A
:
69
70??cM
?#=Q?
2,/
?
^?
^?? i
a
ü ??.
(1)
0
BB
@
1?12 13
12 1 12
1
3
1
2?1
1
CC
A; (2)
0
BB
@
1
9?
8
9?
4
9
89 19?49
49?49 79
1
CC
A.
3,(1)?B??_
d?ê_
,#?
^???.
(2)?Z?
B??_
(
^?ê_
,O

??,#1???.
3,
!x1n?
_
,xTx = 1,
7H = E?2xxT,￡
üH
^?￥???.
￡
ü:?i? ?￥?
@(A+B)T = AT +BT,μ
HT = (E?2xxT)T
= ET?2(xxT)T
= E?2(xT)T(xT)
= E?2xxT
= H:
[H
^?￥.?
HTH = (E?2xxT)(E?2xxT)
= E?2xxT?2xxT +4xxTxxT (xTx = 1)
= E:
5H
^???.
8
¤￡H
^?￥???.
4,
!ADB?
^???,￡
üAB9
^???.
￡
ü:y1A,B
^???,#A?1 = AT,B?1 = BT.
(AB)T(AB) = BTATAB = B?1A?1AB = E:
#AB9
^???.
5,p/
?￥+?′+?_
:
(1)
0
BB
@
2?1 2
5?3 3
1 0?2
1
CC
A; (2)
0
BB
@
1 2 3
2 1 3
3 3 6
1
CC
A; (3)
0
BB
BB
@
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
1
CC
CC
A
.
3,(1)?
jAEj =
flfl
flfl
flfl
flfl
21 2
5?3 3
1 0?2
flfl
flfl
flfl
flfl
c3+(?+2)c1=========
flfl
flfl
flfl
flfl
21?2?2
5?357
1 0 0
flfl
flfl
flfl
flfl
= (?+1)3;
¤A￥+?′1?1 =?2 =?3 =?1.
1 =?2 =?3 =?1
H,3Z?(A+E)x = 0,?
(A+E) =
0
BB
@
3?1 2
5?2 3
1 0?1
1
CC
A
r1 + 3r3
^r
2 + 5r3
0
BB
@
0?1?1
0?2?2
1 0?1
1
CC
A^
0
BB
@
1 0 1
0 1 1
0 0 0
1
CC
A;
L?}
(]6
1?)5? Is?71
¤$3"p =
0
BB
@
1
1
1
1
CC
A.
[kp (k 6= 0)
^?1 =?2 =?3 =?1￥??+?′_
.
(2)?
jAEj =
flfl
flfl
flfl
flfl
1 2 3
2 1 3
3 3 6
flfl
flfl
flfl
flfl
=(?+1)(9);
¤A￥+?′1?1 = 0,?2 =?1,?3 = 9.
1 = 0
H,3Z?Ax = 0,?
A =
0
BB
@
1 2 3
2 1 3
3 3 6
1
CC
A
r2?2r1
^r
3?r1?r2
0
BB
@
1 2 3
0?3?3
0 0 0
1
CC
A
r1 + r2
^r
2 ￥(?3)
0
BB
@
1?1 0
0 1 1
0 0 0
1
CC
A;
¤$3"p1 =
0
BB
@
1
1
1
1
CC
A,#k1p1(k1 6= 0)
^?1 = 0￥??+?′_
.
2 =?1
H,3Z?(A+E)x = 0,?
A+E =
0
BB
@
2 2 3
2 2 3
3 3 7
1
CC
A^
0
BB
@
2 2 3
0 0 1
0 0 0
1
CC
A;
¤$3"p2 =
0
BB
@
1
1
0
1
CC
A,#k2p2(k2 6= 0)
^?2 =?1￥??+?′_
.
3 = 9
H,3Z?(A?9E)x = 0,?
A?9E =
0
BB
@
8 2 3
2?8 3
3 3?3
1
CC
A
r1 + r3
^r
2 + r3
0
BB
@
5 5 0
5?5 0
3 3?3
1
CC
A^
0
BB
@
1 1 0
0 0 0
1 1?1
1
CC
A^
0
BB
@
1 1 0
0 0 0
0 2?1
1
CC
A;
¤$3"p3 =
0
BB
@
1
1
2
1
CC
A,#k3p3(k3 6= 0)
^?3 = 9￥??+?′_
.
(3)?
jAEj =
flfl
flfl
flfl
flfl
flfl
0 0 1
0 1 0
0 1 0
1 0 0
flfl
flfl
flfl
flfl
flfl
Z 7r1=======
flfl
flfl
flfl
flfl
1 0
1 0
0 0
flfl
flfl
flfl
flfl
+1￠(?1)1+4
flfl
flfl
flfl
flfl
0 1
0 1
1 0 0
flfl
flfl
flfl
flfl
=?2(1)?1￠(?2?1) = (?2?1)2
=(?+1)2(1)2;
72??cM
?#=Q?
¤
1 =?2 =?1;?3 =?4 = 1:
1 =?2 =?1
H,3Z?(A+E)x = 0.?
A+E =
0
BB
BB
@
1 0 0 1
0 1 1 0
0 1 1 0
1 0 0 1
1
CC
CC
A^
0
BB
BB
@
1 0 0 1
0 1 1 0
0 0 0 0
0 0 0 0
1
CC
CC
A;
¤$3"
p1 =
0
BB
BB
@
1
0
0
1
1
CC
CC
A; p2 =
0
BB
BB
@
0
1
1
0
1
CC
CC
A
:
[?1 =?2 =?1￥??+?_
1
k1p1 +k2p2 (k1; k2?]
H10):
3 =?4 = 1
H,3Z?(A?E)x = 0.?
A+E =
0
BB
BB
@
1 0 0 1
0?1 1 0
0 1?1 0
1 0 0?1
1
CC
CC
A^
0
BB
BB
@
1 0 0 1
0?1 1 0
0 0 0 0
0 0 0 0
1
CC
CC
A;
¤$3"
p3 =
0
BB
BB
@
1
0
0
1
1
CC
CC
A; p4 =
0
BB
BB
@
0
1
1
0
1
CC
CC
A
:
[?3 =?4 = 1￥??+?_
1
k3p3 +k4p4 (k3; k4?]
H10):
6,
!A1n¨ ?,￡
üATDA￥+?′M].
￡
ü:￡
ü=?μM]￥+?Z?(+?[
T)' V.??éflflATflfl = jAj,?
flflATEflfl = flfl(AE)Tflfl = jAEj:
¤￡ATDA￥+?′M].
7,
!n¨ ?AaB
@R(A)+R(B) < n,￡
üADBμ
￥+?′,μ
￥+?_
.
￡
ü:?R(A)+R(B) < n,μR(A) < n,7
R(A) < n,jAj = 0,jA?0Ej = 0,0
^A￥+?′:
] ?,09
^B￥+?′.
[ADBμ
￥+?′0.
/￡ADBμ? = 0￥
+?_
.
ADBμ? = 0￥
+?_

L?}
(]6
1?)5? Is?73
,id
,_
p]
H
@Ap = 0p; Bp = 0p
,Z?F
8<
:
Ax = 0
Bx = 0
μd
,3
,Z?F
A
B
!
x = 0μd
,3
7
R
A
B
!
= R(AT; BT) 6 R(AT)+R(BT) = R(A+R(B) < n:
8
?ADBμ
￥+?_
.
8,
!A2?3A+2E = O,￡
üA￥+?′o
|12.
￡
ü:
!?
^A￥+?′,5id
,_
p
P
Ap =?p:
H]
HPeA,μ
A2p =?Ap =?2p,(5.1)
??A2?3A+2E = O,¤A2 = 3A?2E,
[
A2p = (3A?2E)p = 3?p?2p,(5.2)
8(5.1)
T(5.2)
T¤
2p = 3?p?2p;'(?2?3?+2)p = 0:
7+?_
p 6= 0,
[
2?3?+2 = 0:
3¤
= 12:
¤￡A￥+?′o
|12.
6￡.
!?
^A￥+?′,5?2?3?+2
^A2?3A+2E￥+?′1.5id
,_
p
P
(A2?3A+2E)p = (?2?3?+2)p:
??A2?3A+2E = O,}?

T¤
(?2?3?+2)p = 0:
7+?_
p 6= 0,
[
2?3?+2 = 0:
3¤? = 12.
¤￡A￥+?′o
|12.
B?μ?J￥￡
ü:
?A2?3A+2E = O,¤(A?2E)(A?E) = O.
H|?

T¤
j(A?2E)(A?E)j = jA?2EjjA?Ej = 0:
1nP.122 è8￥w<2
.
74??cM
?#=Q?
[
jA?2Ej = 0jA?Ej = 0;
512
^ ?A￥+?′.
?
^?"o
^
a
ü
12
^ ?A￥+?′, ?A
^?μY￥+?′
àμ¤?￡
ü,?ü?
/2
a\A￥+?′.o.
.|12".
9,
!A1???,OjAj =?1,￡
ü? =?1
^A￥+?′.
￡
ü:'3￡
ü? =?1
@+?Z?jAEj = 0,'jA+Ej = 0.y1
jA+Ej = flflA+ATAflfl (A1???)
= flflE +ATflfljAj
=?flflAT +Eflfl (jAj =?1)
=?flfl(A+E)Tflfl
=?jA+Ej;
[2jA+Ej = 0,'jA+Ej = 0.¤￡? =?1
^A￥+?′.
10,
!? 6= 0
^m¨ ?Am￡nBn￡m￥+?′,￡
ü?9
^n¨ ?BA￥+?′.
￡
ü:
!p
^ ?Am￡nBn￡m￥?￥+?_
,5
(AB)p =?p,(5.3)

T
H]
HPeB¤B(AB)p = B?p,'
(BA)(Bp) =?(Bp):
/
￡
üBp
^d
,￥.y1,L?Bp = 0,5(5.3)
T?PH(AB)p = A(Bp) = 0;?
^? 6= 0,O
+?_
p
^d
,_
,V7?p 6= 0.L
!?? ?.
¤￡?9
^n¨ ?BA￥+?′.
?i:+?_

^d
,￥.
11,X?3¨ ?A￥+?′11,2,3,pflflA3?5A2 +7Aflfl.
3,(
_P.122 è935.)
!?
^ ?A￥+?′,:’(A) = A3?5A2+7A,5’(?) =?3?5?2+7?
^’(A)￥+?′.?
’(1) = 3; ’(2) = 2; ’(3) = 3;
?’(A)￥+?′13,2,3,
[
flflA3?5A2 +7Aflfl = 3￡2￡3 = 18:
12,X?3¨ ?A￥+?′11,2,?3,pjA? +3A+2Ej.
3:yA￥+?′??10,?A V
I,#A? = jAjA?1.7jAj =?1?2?3 =?6,
[
A? +3A+2E =?6A?1 +3A+2E:
ü

T:T’(A),μ’(?) =?6? +3?+2,#’(A)￥+?′1
’(1) =?1; ’(2) = 5; ’(?3) =?5;
?
^
jA? +3A+2Ej = (?1)￠5￠(?5) = 25:
L?}
(]6
1?)5? Is?75
63.?i?jAj =?6,59
jAjjA? +3A+2Ej,'flfl?6E +3A2 +2Aflfl.^¤?6E + 3A2 + 2A￥
+?′1?1,10,15,
[fl
fl?6E +3A2 +2Aflfl = (?1)￠10￠15 =?150;
V7¤jA? +3A+2Ej = 25.
13,
!A,B?
^n¨Z?,OA V
I,￡
üABDBAM
.
￡
ü:?A V
I?
A?1(AB)A = (A?1A)(BA) = BA;
5ABDBAM
.
14,
! ?A =
0
BB
@
2 0 1
3 1 x
4 0 5
1
CC
A VM
?,px.
3:35G :? ?4 (P.125),\n¨ ?A
?￥ 1Hq
^Aμn?L?í1￥+?_
".
jAEj =
flfl
flfl
flfl
flfl
2 0 1
3 1 x
4 0 5
flfl
flfl
flfl
flfl
=(1)
flfl
flfl
fl
2 1
4 5
flfl
flfl
fl =?(1)
2(6);
¤?1 = 6,?2 =?3 = 1.
1
P3¨ ?A
?,53Aμ3?L?í1￥+?_
.
??1 = 1, Vp¤L?í1￥+?_
μ1?,
[,1
P ?A
?,3?×?
2 =?3 = 1μ2?L?í1￥+?_
,'Z?(A?E)x = 0μ2?L?í1￥3.?P.98? ?7,'
R(A?E) = 1:
?
A?E =
0
BB
@
1 0 1
3 0 x
4 0 4
1
CC
A^
0
BB
@
1 0 1
0 0 x?3
0 0 0
1
CC
A;
¤
x = 3:
yN,?x = 3
H, ?A
?.
15,X?p =
0
BB
@
1
1
1
1
CC
A
^ ?A =
0
BB
@
2?1 2
5 a 3
1 b?2
1
CC
A￥B?+?_
.
(1)p?
a,b#+?_
p
?￥+?′;
(2)ùA
?
M
i
a
ü ??.
3,(1)
!+?_
p
?￥+?′1?,5Ap =?p,'
Ap =
0
BB
@
2?1 2
5 a 3
1 b?2
1
CC
A
0
BB
@
1
1
1
1
CC
A =
0
BB
@
1
2+a
1+b
1
CC
A =?
0
BB
@
1
1
1
1
CC
A;
76??cM
?#=Q?
¤
=?1; a =?3; b = 0:
(2)?
jAEj =
flfl
flfl
flfl
flfl
21 2
5?3 3
1 0?2
flfl
flfl
flfl
flfl
=?(?+1)3;
¤
1 =?2 =?3 =?1:
1
P3¨ ?A
?,3
P×1 =?2 =?3 =?1?3?L?í1￥+?_
,??
^? V
￥.y13í
QZ?(A+E)x = 0￥L?í13￥?
1
n?r = 3?R(A+E);
7? úA+E 6= O,'R(A+E) > 1,
[,Z?(A+E)x = 0￥L?í13￥?
? V
13.
yN, ?A?
M
?.
16,
kpB???￥M
MD ?,|/
? ?1? ?:
(1)
0
BB
@
2?2 0
2 1?2
0?2 0
1
CC
A; (2)
0
BB
@
2 2?2
2 5?4
2?4 5
1
CC
A.
3,(1)?
jAEj =
flfl
flfl
flfl
flfl
22 0
2 12
0?2
flfl
flfl
flfl
flfl
= (1)(4)(?+2);
¤ ?jAj￥+?′1?1 =?2,?2 = 1,?3 = 4.
1 =?2
H,?0
BB
@
4?2 0
2 3?2
0?2 2
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A = 0;
3¤0
BB
@
x1
x2
x3
1
CC
A = k1
0
BB
@
1
2
2
1
CC
A:
?ê+?_
V|1p1 = 13
0
BB
@
1
2
2
1
CC
A.
2 = 1
H,?0
BB
@
1?2 0
2 0?2
0?2?1
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A = 0;
3¤0
BB
@
x1
x2
x3
1
CC
A = k2
0
BB
@
2
1
2
1
CC
A:
L?}
(]6
1?)5? Is?77
?ê+?_
V|1p2 = 13
0
BB
@
2
1
2
1
CC
A.
3 = 4
H,?0
BB
@
2?2 0
2?3?2
0?2?4
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A = 0;
3¤0
BB
@
x1
x2
x3
1
CC
A = k3
0
BB
@
2
2
1
1
CC
A:
?ê+?_
V|1p3 = 13
0
BB
@
2
2
1
1
CC
A.
¤???(p1;p2;p3) = P = 13
0
BB
@
1 2 2
2 1?2
2?2 1
1
CC
A,
[ ?A V?1
P?1AP =
0
BB
@
2 0 0
0 1 0
0 0 4
1
CC
A:
(2)?
jAEj =
flfl
flfl
flfl
flfl
2 2?2
2 54
2?4 5
flfl
flfl
flfl
flfl
=?(1)2(10);
¤+?′1?1 =?2 = 1,?3 = 10.
1 =?2 = 1
H,?0
BB
@
1 2?2
2 4?4
2?4 4
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
0
0
0
1
CC
A;
3¤0
BB
@
x1
x2
x3
1
CC
A = k1
0
BB
@
2
1
0
1
CC
A+k2
0
BB
@
2
0
1
1
CC
A:
'$3"1?1 = (?2; 1; 0)T,?2 = (2; 0; 1)T.|?1,?2??,|·1 =?1,
·2 =?2? [·1;?2][·
1; ·1]
·1 =
0
BB
@
2
0
1
1
CC
A?
4
5
0
BB
@
2
1
0
1
CC
A =
0
BB
@
2=5
4=5
1
1
CC
A:
?ê¤
p1 = ·1k·
1k
= 1p5
0
BB
@
2
1
0
1
CC
A; p2 =
1
3p5
0
BB
@
2
4
5
1
CC
A:
78??cM
?#=Q?
3 = 10
H,?0
BB
@
8 2?2
2?5?4
2?4?5
1
CC
A
0
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
0
0
0
1
CC
A;
3¤0
BB
@
x1
x2
x3
1
CC
A = k3
0
BB
@
1
2
2
1
CC
A:
?ê¤p3 = 13
0
BB
@
1
2
2
1
CC
A.
¤???
(p1; p2; p3) =
0
BB
@
2p5 2
p5
15?
1
3
1p
5
4p5
15?
2
3
0
p5
3
2
3
1
CC
A:
V7 ?A V?1
P?1AP =
0
BB
@
1 0 0
0 1 0
0 0 10
1
CC
A:
17,
! ?A =
0
BB
@
1?2?4
2 x?2
4?2 1
1
CC
AD? =
0
BB
@
5
4
y
1
CC
AM
,px,y;ipB????P,
P
P?1AP =?.
3:Z?AD?M
,55,?4,y
^ ?A￥+?′.V7jA?5Ej = 0,jA+4Ej = 0.?
jA+4Ej = 0¤fl
flfl
flfl
flfl
fl
5?2?4
2 x+4?2
4?2 5
flfl
flfl
flfl
flfl
= 0;
9
¤
x = 4:
/py.??éjAj =?1?2?3,¤
jAj =
flfl
flfl
flfl
flfl
1?2?4
2 4?2
4?2 1
flfl
flfl
flfl
flfl
= 5￡(?4)￡y;
9
¤
y = 5:
/p?? ?P.
1 =?2 = 5
H,3Z?(A?5E)x = 0.?
A?5E =
0
BB
@
4?2?4
2?1?2
4?2?4
1
CC
A^
0
BB
@
2 1 2
0 0 0
0 0 0
1
CC
A;
L?}
(]6
1?)5? Is?79
¤$3"
1 = (1;?2; 0)T;?2 = (0;?2; 1)T:
S??,¤
·1 = 1p5(1;?2; 0)T; ·2 = 13p5(4; 2;?5)T:
3 =?4
H,3Z?(A+4E)x = 0.?
A+4E =
0
BB
@
5?2?4
2 8?2
4?2 5
1
CC
A^
0
BB
@
1?2 0
0?2 1
0 0 0
1
CC
A;
¤?ê$3"
·3 = 13(2; 1; 2)T:
7P = (·1; ·3; ·2),5P1
p￥B??? ?,i
@
P?1AP =?:
(?iP?+?_
￥

¨?,
^D??￥?íM?￥.)
63(px,y).Z?AD?M
,55,?4,y
^ ?A￥+?′.?+?′?é?1 +?2 +?3 =
a11 +a22 +a33jAj =?1?2?3 V¤
8
<
:
1 +?2 +?3 = 1+x+1;
1?2?3 = jAj:
'8
<
:
1+y = 2+x;
20y =?15x?40:
3¤8
<
:
x = 4;
y = 5:
18,
!3¨ ?A￥+?′1?1 = 2,?2 =?2,?3 = 1;?￥+?_
GQ1
p1 =
0
BB
@
0
1
1
1
CC
A; p2 =
0
BB
@
1
1
1
1
CC
A; p3 =
0
BB
@
1
1
0
1
CC
A;
pA.
3::P = (p1; p2p3),? = diag(?1;?2;?3),5
P?1AP =?:
[
A = P?P?1 =
0
BB
@
0 1 1
1 1 1
1 1 0
1
CC
A
0
BB
@
2
2
1
1
CC
A
0
BB
@
0 1 1
1 1 1
1 1 0
1
CC
A
1
=
0
BB
@
0?2 1
2?2 1
2?2 0
1
CC
A
0
BB
@
0 1 1
1 1 1
1 1 0
1
CC
A
1
:
80??cM
?#=Q?
?
0
BB
BB
BB
BB
B@
0 1 1
1 1 1
1 1 0
0?2 1
2?2 1
2?2 0
1
CC
CC
CC
CC
CA
c2?c1
^
0
BB
BB
BB
BB
B@
0 1 1
1 0 1
1 0 0
0?2 1
2?4 1
2?4 0
1
CC
CC
CC
CC
CA
c3?c2
^
0
BB
BB
BB
BB
B@
0 1 0
1 0 1
1 0 0
0?2 3
2?4 5
2?4 4
1
CC
CC
CC
CC
CA
c1?c3
^
0
BB
BB
BB
BB
B@
0 1 0
0 0 1
1 0 0
3?2 3
3?4 5
2?4 4
1
CC
CC
CC
CC
CA
^
0
BB
BB
BB
BB
B@
1 0 0
0 1 0
0 0 1
2 3?3
4 5?3
4 4?2
1
CC
CC
CC
CC
CA
[
A = P?P?1 =
0
BB
@
2 3?3
4 5?3
4 4?2
1
CC
A:
19,
!3¨? ?A￥+?′1?1 = 1,?2 =?1,?3 = 0;1,?2￥+?_
GQ1
p1 =
0
BB
@
1
2
2
1
CC
A; p2 =
0
BB
@
2
1
2
1
CC
A:
pA.
3:
!?3?￥+?_
1p3 = (x; y; z)T.?i?A1??,1 6=?2 6=?3,?p1,p2,p3??.
58
<
:
x+2y +2z = 0;
2x+y?2z = 0:
(5.4)
?"
?
A =
1 2 2
2 1?2
!
r2 + r1
^
1 2 2
3 3 0
!
^
1 2 2
1 1 0
!
^
0 1 2
1 1 0
!;
?Z?F(5.4)￥Y31
(x; y; z)T = k(?2; 2;?1)T:
V|
p3 = (?2; 2;?1)T;
yA?,Aμ???Q,
P
QTAQ = Q?1AQ = diag(1;?1; 0):
-
Xüp¤p1,p2,p3??,?ê,'¤
Q =
0
BB
@
1=3 2=3?2=3
2=3 1=3 2=3
2=3?2=3?1=3
1
CC
A:
[
A = Q diag(1;?1; 0)QT = 19
0
BB
@
1 2?2
2 1 2
2?2?1
1
CC
A
0
BB
@
1
1
0
1
CC
A
0
BB
@
1 2 2
2 1?2
2 2?1
1
CC
A =
1
3
0
BB
@
1 0 2
0 1 2
2 2 0
1
CC
A:
L?}
(]6
1?)5? Is?81
20,
!3¨? ?A￥+?′1?1 = 6,?2 =?3 = 3,D+?′?1 = 6?￥+?_
1
p1 = (1; 1; 1)T,pA.
3,1–5p?2,?3
?￥+?_
p2,p3.?? ?6,p2,p3Dp1??.
!(x; y; z)TDp1??,
5
x+y +z = 0:
3Z?¤$3"1
(?1; 0; 1)T; (?1; 1; 0)T:
[ V|
p2 = (?1; 0; 1)T; p3 = (?1; 1; 0)T:
2–/p ?A.?A
^??,i???Q
P¤QTAQ = Q?1AQ = diag(6; 3; 3).
[ V[?p1,
p2,p3¤?BF?ê??_
,[????Q.
|p2,p3??:|·2 = p2,
·3 = p3? [p3; ·2][·
2; ·2]
·2 = 12
0
BB
@
1
2
1
1
CC
A:
|p1,·2,·3?ê,¤
q1 = 1p3
0
BB
@
1
1
1
1
CC
A; q2 =
1p
2
0
BB
@
1
0
1
1
CC
A; q3 =
1p
6
0
BB
@
1
2
1
1
CC
A:
7Q = (q1; q2; q3),5Q1???,¤
A = Q diag(6; 3; 3)QT =
0
BB
@
4 1 1
1 4 1
1 1 4
1
CC
A:
21,
!a = (a1; a2; ￠￠￠ ; an)T,a1 6= 0,A = aaT,
(1)￡
ü? = 0
^A￥n?1×+?′;
(2)pA￥d
,+?′#n?L?í1￥+?_
.
3,(1)?i?A1??,#AD? = diag(?1;?2; ￠￠￠ ;?n)M
,1,?2,￠￠￠,?n
^A
￥??+?′.
?5?185,?R(A) = 1,V7R(?) = 1,?
^?￥?íoμB?d
,,'? = 0
^A￥n?1
×+?′.
(2)yA = aaT￥?Lí
í-1
nX
i=1
a2i;??+?′?é,A￥n?+?′-1
nX
i=1
a2i,X￡
= 0
^A￥n?1×+?′,
[
:/￥
*?+?′o
^
nX
i=1
a2i.
X?a1 6= 0,
[
nX
i=1
a2i 6= 0,¤￡
nX
i=1
a2i
^A￥d
,+?′(O
^?B￥).
/pA￥+?_
.
(a) = 0
H,p3Z?Ax = 0.?
A =
0
BB
BB
B@
a21 a1a2 ￠￠￠ a1an
a2a1 a22 ￠￠￠ a2an
...,..,..
ana1 ana2 ￠￠￠ a2n
1
CC
CC
CA
r1 ￥a1
^
0
BB
BB
B@
a1 a2 ￠￠￠ an
a2a1 a22 ￠￠￠ a2an
...,..,..
ana1 ana2 ￠￠￠ a2n
1
CC
CC
CA
ri?air1
^i = 2;3;￠￠￠
0
BB
BB
B@
a1 a2 ￠￠￠ an
0 0 ￠￠￠ 0
...,..,..
0 0 ￠￠￠ 0
1
CC
CC
CA;
82??cM
?#=Q?
¤? = 0?￥??+?_
1
x = k1
0
BB
BB
BB
B@
a2=a1
1
0
...
0
1
CC
CC
CC
CA
+k2
0
BB
BB
BB
B@
a3=a1
0
1
...
0
1
CC
CC
CC
CA
+￠￠￠+kn?1
0
BB
BB
BB
B@
an=a1
0
0
...
1
1
CC
CC
CC
CA
(k1; k2; ￠￠￠ ; kn?1?]
H10):
(b)1 =
nX
i=1
a2i
H,?A = aaT,μ
Aa = (aaT)a = a(aTa) = a
nX
i=1
a2i =
nX
i=1
a2i
·
a;
Vn?1 =
nX
i=1
a2i?￥+?′ü
^a.
22,
!A =
0
BB
@
1 4 2
0?3 4
0 4 3
1
CC
A,pA100.
3,(B?￥3E)5ü ?A?.?
jAEj =
flfl
flfl
flfl
flfl
1 4 2
0?3 4
0 4 3
flfl
flfl
flfl
flfl
= (1)(5)(?+5);
¤A￥+?′1?1 = 1,?2 = 5,?3 =?5.
1 = 1
H,3Z?(A?E)x = 0.?
A?E =
0
BB
@
0 4 2
0?4 4
0 4 2
1
CC
A^
0
BB
@
0 1 0
0 0 1
0 0 0
1
CC
A;
V|?1 = 1?￥+?_
1p1 = (1; 0; 0)T.
2 = 5
H,3Z?(A?5E)x = 0.?
A?5E =
0
BB
@
4 4 2
0?8 4
0 4?2
1
CC
A^
0
BB
@
1 0?1
0 2?1
0 0 0
1
CC
A;
V|?2 = 5?￥+?_
1p2 = (2; 1; 2)T.
3 =?5
H,3Z?(A?5E)x = 0.?
A?5E =
0
BB
@
6 4 2
0 2 4
0 4 8
1
CC
A^
0
BB
@
1 0?1
0 1 2
0 0 0
1
CC
A;
V|?3 =?5?￥+?_
1p3 = (1;?2; 1)T.
:P = (p1; p2; p3),? = diag(1; 5;?5),5P?1AP =?.
[
A = P?P?1:
L?}
(]6
1?)5? Is?83
5
A100 = P?100P?1 =
0
BB
@
1 2 1
0 1?2
0 2 1
1
CC
A
0
BB
@
1 0 0
0 5100 0
0 0 5100
1
CC
A
0
BB
@
1 2 1
0 1?2
0 2 1
1
CC
A
1
=
0
BB
@
1 2￠5100 5100
0 5100?2￠5100
0 2￠5100 5100
1
CC
A
0
BB
@
1 2 1
0 1?2
0 2 1
1
CC
A
1
=
0
BB
@
1 0 5100?1
0 5100 0
0 0 5100
1
CC
A:
63,(B?
·F￥3E)?
0
BB
@
1 4 2
0?3 4
0 4 3
1
CC
A
0
BB
@
1 4 2
0?3 4
0 4 3
1
CC
A =
0
BB
@
1 0 24
0 25 0
0 0 25
1
CC
A;
70
BB
@
a 0 d
0 b 0
0 0 c
1
CC
A
0
BB
@
a 0 d
0 b 0
0 0 c
1
CC
A =
0
BB
@
a2 0 (a+c)d
0 b2 0
0 0 c2
1
CC
A;
?d = c?a,5
0
BB
@
a 0 c?a
0 b 0
0 0 c
1
CC
A
0
BB
@
a 0 c?a
0 b 0
0 0 c
1
CC
A =
0
BB
@
a2 0 c2?a2
0 b2 0
0 0 c2
1
CC
A
￠￠￠￠￠￠
0
BB
@
an 0 cn?an
0 bn 0
0 0 cn
1
CC
A
0
BB
@
a 0 c?a
0 b 0
0 0 c
1
CC
A =
0
BB
@
an?1 0 cn?1?an?1
0 bn?1 0
0 0 cn?1
1
CC
A
[
A100 =
0
BB
@
1 0 24
0 25 0
0 0 25
1
CC
A
50
=
0
BB
@
1 0 2550?1
0 2550 0
0 0 2550
1
CC
A =
0
BB
@
1 0 5100?1
0 5100 0
0 0 5100
1
CC
A:
???5?×1.3N ??￥5
"￥
H,?1B?1oX?
P¨?￥ZE,1
21G 5
"
￥+?p3.1?/
￥5
".
!A =
0
BB
@
1 1 1
1?1?1
1?1?1
1
CC
A,9
A5.
3EB?i?
A =
0
BB
@
1 1 1
1?1?1
1?1?1
1
CC
A =
0
BB
@
1
1
1
1
CC
A
1?1?1
·;
5
A5 =
0
BB
@
1
1
1
1
CC
A
1?1?1
·
0
BB
@
1
1
1
1
CC
A
1?1?1
·
￠￠￠
0
BB
@
1
1
1
1
CC
A
1?1?1
·
:
84??cM
?#=Q?
7
1?1?1
·
0
BB
@
1
1
1
1
CC
A =?3;
[
A5 =
0
BB
@
1
1
1
1
CC
A￠(?3)4 ￠
1?1?1
·
= (?3)4
0
BB
@
1
1
1
1
CC
A
1?1?1
·
= 81A =
0
BB
@
81 81 81
81?81?81
81?81?81
1
CC
A:
3E=?
A2 =
0
BB
@
1 1 1
1?1?1
1?1?1
1
CC
A
0
BB
@
1 1 1
1?1?1
1?1?1
1
CC
A =
0
BB
@
3?3?3
3 3 3
3 3 3
1
CC
A =?3A;
V?
A5 = (?3)4A =
0
BB
@
81 81 81
81?81?81
81?81?81
1
CC
A:
AB?5
".
X?A =
0
BB
@
a a a
b b b
c c c
1
CC
A,pA2006.
3:?i?A =
0
BB
@
a
b
c
1
CC
A(1; 1; 1),5
A2006 =
0
BB
@
a
b
c
1
CC
A(1; 1; 1)
0
BB
@
a
b
c
1
CC
A(1; 1; 1)￠￠￠
0
BB
@
a
b
c
1
CC
A(1; 1; 1)
| {z }
2006?A
=
0
BB
@
a
b
c
1
CC
A(1; 1; 1)
0
BB
@
a
b
c
1
CC
A(1; 1; 1)￠￠￠
0
BB
@
a
b
c
1
CC
A
| {z }
2005?(a+b+c)Me
(1; 1; 1)
= (a+b+c)2005
0
BB
@
a a a
b b b
c c c
1
CC
A:
23,
S,
Mμ1 è1p￥
jh
M ì?,μ1 è1q￥ì?
M
jh.L
!?S9|
g
?M,O

| g%M￥
p9?M.ün
Ma
jh| gì?| g]9| g￥1 èGQ:1xnyn
(xn +yn = 1).
(1)p1"
T
x
n+1
yn+1
= A
x
n
yn;
(2)
!
"-
jh| gDì?| gM?,'
x
0
y0
=
0:5
0:5
,p
x
n
yn
.
L?}
(]6
1?)5? Is?85
3:?5
!¤
8<
:
xn+1 = (1?p)xn +qyn;
yn+1 = pxn +(1?q)yn:
'
xn+1
yn+1
!
=
1?p q
p 1?q
!?
xn
yn
!;
[
A =
1?p q
p 1?q
!
:
(2)??w1"
T?
xn
yn
!
= A
xn?1
yn?1
!
= ￠￠￠ = An
x0
y0
!;
}?
x
0
y0
=
0:5
0:5
,5
xn
yn
!
= 0:5An
1
1
!;
/pAn.?
AE =
1?p q
p 1?q
!
=1￠￡(1?p?q)?;
¤ ?A￥+?′1?1 = 1,?2 = 1?p?q.
1 = 1
H,3Z?(A?E)x = 0.?
A?E =
p q
p?q
!
^
p q
0 0
!;
V|?1 = 1
?￥+?_
1?1 =
q
p
.
2 = 1?p?q
H,3ZA?(1?p?q)E￠x = 0.?
A?(1?p?q)E =
q q
p p
!
^
1 1
0 0
!;
V|?2 = 1?p?q
?￥+?_
1?2 =
1
1
.
7P = (?1;?2),5P V
I,OP?1AP = diag(1; 1?p?q).
[A = P diag(1; 1?p?q)P?1,¤
xn
yn
!
= An
x0
y0
!
= P
1 0
0 (1?p?q)n
!
P?1
x0
y0
!
= 12(p+q)
2q?(q?p)(1?p?q)n
2p+(q?p)(1?p?q)n
!
:
24,(1)
!A =
3?2
2 3
!
,p’(A) = A10?5A9;
(2)
!A =
0
BB
@
2 1 2
1 2 2
2 2 1
1
CC
A,p’(A) = A10?6A9 +5A8.
3,(1) ?A =
3?2
2 3
!
^
L? ?, Vs???M
MD ?P = 1p2
1?1
1 1
!
,
P¤
P?1AP =
1 0
0 5
!
=?:
86??cM
?#=Q?
V7
A = P?P?1; Ak = P?kP?1:
yN
’(A) = A10?5A9
= P?10P?1?5P?9P?1
= P
1 0
0 510
!
P?1?P
5 0
0 510
!
P?1
= P
4 0
0 0
!
P?1
= 1p2
1?1
1 1
!?
4 0
0 0
!
1p
2
1 1
1 1
!
=
2?2
2?2
!
:
(2)](1)p¤??M
MD ?
P =
0
BB
@
p6
6?
1p
2
1p
3
p6
6
1p
2
1p
3p
6
3 0
1p
3
1
CC
A;
P¤
P?1AP =
0
BB
@
1 0 0
0 1 0
0 0 5
1
CC
A =?;
V7
A = P?P?1; Ak = P?kP?1:
yN
’(A) = A10?6A9 +5A8
= A8(A2?6A+5E)
= A8(A?E)(A?5E)
= P?8P?1
0
BB
@
1 1 2
1 1 2
2 2 0
1
CC
A
0
BB
@
3 1 2
1?3 2
2 2?4
1
CC
A
= 2
0
BB
@
1 1?2
1 1?2
2?2 4
1
CC
A:
25,¨ ?:|V
U/
=Q?:
(1) f = x2 +4xy +4y2 +2xz +z2 +4yz;
(2) f = x2 +y2?7z2?2xy?4xz?4yz;
(3) f = x21 +x22 +x23 +x24?2x1x2 +4x1x3?2x1x4 +6x2x3?4x2x4:
L?}
(]6
1?)5? Is?87
3,(1) f = (x; y; z)
0
BB
@
1 2 1
2 4 2
1 2 1
1
CC
A
0
BB
@
x
y
z
1
CC
A.
(2) f = (x; y; z)
0
BB
@
1?1?2
1 1?2
2?2?7
1
CC
A
0
BB
@
x
y
z
1
CC
A.
(3) f = (x1; x2; x3; x4)
0
BB
BB
@
1?1 2?1
1 1 3?2
2 3 1 0
1?2 0 1
1
CC
CC
A
0
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
.
26,/
=Q?￥ ?:
(1) f(x) = xT
2 1
3 1
!
x; (2) f(x) = xT
0
BB
@
1 2 3
4 5 6
7 8 9
1
CC
Ax.
3:?1??aijDaji￥′,
P
?￥?M,Oaij = aji' V.
(1) A =
2 2
2 1
!
.
(2) A =
0
BB
@
1 3 5
3 5 7
5 7 9
1
CC
A.
27,pB???MD|/
=Q??S?:
(1) f = 2x21 +3x22 +3x23 +4x2x3;
(2) f = x21 +x22 +x23 +x24 +2x1x2?2x1x4?2x2x3 +2x3x4.
3,(1)=Q?f￥ ?1A =
0
BB
@
2 0 0
0 3 2
0 2 3
1
CC
A.?
jAEj =
flfl
flfl
flfl
flfl
2 0 0
0 3 2
0 2 3
flfl
flfl
flfl
flfl
= (2)(5)(1);
¤A￥+?′1?1 = 2,?2 = 5,?3 = 1.
1 = 2
H,3Z?(A?2E)x = 0,?
A?2E =
0
BB
@
0 0 0
0 1 2
0 2 1
1
CC
A?
0
BB
@
0 1 2
0 0 1
0 0 0
1
CC
A:
¤$3"?1 =
0
BB
@
1
0
0
1
CC
A,|p1 =
0
BB
@
1
0
0
1
CC
A.
2 = 5
H,3Z?(A?5E)x = 0,?
A?5E =
0
BB
@
3 0 0
0?2 2
0 2?2
1
CC
A?
0
BB
@
1 0 0
0 1?1
0 0 0
1
CC
A:
88??cM
?#=Q?
¤$3"?2 =
0
BB
@
0
1
1
1
CC
A,|p2 =
1p
2
0
BB
@
0
1
1
1
CC
A.
3 = 1
H,3Z?(A?E)x = 0,?
A?E =
0
BB
@
1 0 0
0 2 2
0 2 2
1
CC
A?
0
BB
@
1 0 0
0 1 1
0 0 0
1
CC
A;
¤$3"?3 =
0
BB
@
0
1
1
1
CC
A,|p3 =
1p
2
0
BB
@
0
1
1
1
CC
A,
?
^??MD10
BB
@
x1
x2
x3
1
CC
A =
0
BB
@
1 0 0
0 1=p2?1=p2
0 1=p2 1=p2
1
CC
A
0
BB
@
y1
y2
y3
1
CC
A;
Oμ
f = 2y21 +5y22 +y23:
(2)=Q?f￥ ?1A =
0
BB
BB
@
1 1 0?1
1 1?1 0
0?1 1 1
1 0 1 1
1
CC
CC
A
.?
jAEj =
flfl
flfl
flfl
flfl
flfl
1 1 0?1
1 11 0
0?1 1 1
1 0 1 1
flfl
flfl
flfl
flfl
flfl
= (?+1)(3)(1)2;
¤A￥+?′1?1 =?1,?2 = 3,?3 =?4 = 1.
1 =?1
H, V¤?ê+?_
p1 =
0
BB
BB
@
1
2
12
12
1
2
1
CC
CC
A
,
2 = 3
H, V¤?ê+?_
p2 =
0
BB
BB
@
1
2
1
2
12
12
1
CC
CC
A
,
3 =?4 = 1
H, V¤?ê+?_
p3 =
0
BB
BB
@
1p
2
0
1p
2
0
1
CC
CC
A
,p4 =
0
BB
BB
@
0
1p
2
0
1p
2
1
CC
CC
A
.
?
^??MD10
BB
BB
@
x1
x2
x3
x4
1
CC
CC
A
=
0
BB
BB
B@
1
2
1
2
1p
2 0
12 12 0 1p2
12?12 1p2 0
1
2?
1
2 0
1p
2
1
CC
CC
CA
0
BB
BB
@
y1
y2
y3
y4
1
CC
CC
A
L?}
(]6
1?)5? Is?89
Oμ
f =?y21 +3y22 +y23 +y24:
28,pB???MDü=Qw
￥Z?
3x2 +5y2 +5z2 +4xy?4xz?10yz = 1
?SZ?.
3:ü?
TPH￥=Q?1S?' V.:
A =
0
BB
@
3 2?2
2 5?5
2?5 5
1
CC
A:
?
jAEj =
flfl
flfl
flfl
flfl
3 2?2
2 55
2?5 5
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
3 2?2
2 55
0
flfl
flfl
flfl
flfl
=
flfl
flfl
flfl
flfl
3 4?2
2 105
0 0
flfl
flfl
flfl
flfl
=(2)(11);
¤A￥+?′1?1 = 0,?2 = 2,?3 = 11./p
ì?￥+?_
.
1 = 0
H,?
AE = A =
0
BB
@
3 2?2
2 5?5
2?5 5
1
CC
A
r3 + r2
^
0
BB
@
3 2?2
2 5?5
0 0 0
1
CC
A^
0
BB
@
3 2?2
0 1?1
0 0 0
1
CC
A
r1?2r2
^
0
BB
@
3 0 0
0 1?1
0 0 0
1
CC
A;
V|?1 = 0?￥+?_
1?1 = (0; 1; 1)T.?ê¤p1 = 1p2(0; 1; 1)T.
2 = 2
H,?
A?2E =
0
BB
@
1 2?2
2 3?5
2?5 3
1
CC
A
r3 + r2
^
0
BB
@
1 2?2
2 3?5
0?2?2
1
CC
A
r2?2r1
^
0
BB
@
1 2?2
0?1?1
0?2?2
1
CC
A
r1 + 2r2
^
0
BB
@
1 0 4
0 1 1
0 0 0
1
CC
A;
V|?2 = 2?￥+?_
1?2 = (4;?1; 1)T.?ê¤p2 = 13p2(4;?1; 1)T.
3 = 11
H,?
A?11E =
0
BB
@
8 2?2
2?6?5
2?5?6
1
CC
A
r3 + r2
^
0
BB
@
4 1?1
2?6?5
0?11?11
1
CC
A
r1 + 2r2
^
0
BB
@
0?11?11
2?6?5
0 1 1
1
CC
A
r2 + 5r3
^
0
BB
@
0 0 0
2?1 0
0 1 1
1
CC
A;
V|?3 = 11?￥+?_
1?3 = (1; 2;?2)T.?ê¤p3 = 13(1; 2;?2)T.
V7¤?? ?
P = (p1; p2; p3) =
0
BB
@
0 43p2 13
1p
2?
1
3p2
2
3
1p
2
1
3p2?
2
3
1
CC
A:
???MD(x; y; z)T = P(u; v; w)T,=Q?3x2 +5y2 +5z2 +4xy?4xz?10yz1S?
2v2 +11w2:
90??cM
?#=Q?
'=Qw
￥SZ?1
2v2 +11w2 = 1:
29,￡
ü:=Q?f = xTAxkxk = 1
H￥Kv′1 ?A￥Kv+?′.
￡
ü:?i?A1
L? ?,5i?? ?P,
P¤
PAP?1 =
0
BB
BB
B@
1
2
...
n
1
CC
CC
CA,?:
1;?2;￠￠￠ ;?n1A￥+?′,?^
!?1Kv.
?P1?? ?,5P?1 = PT,OjPj = 1,
[A = P?1?P = PT?P.5
f = xTAx
= xTPT?Px
= yT?y =?1y21 +?2y22 +￠￠￠+?ny2n:
? úy = Px.
?kyk = kPxk = jPjkxk = kxk = 1,'y21 +y22 +￠￠￠+y2n = 1
H,
f = (?1y21 +?2y22 +￠￠￠+?ny2n) 6 (?1y21 +?1y22 +￠￠￠+?1y2n) =?1(y21 +y22 +￠￠￠+y2n) =?1:
#¤￡
max
kxk=1
f = maxf?1;?2; ￠￠￠ ;?ng:
30,¨
￥ZE/
=Q??S?,i
¨MD￥ ?:
(1) f(x1; x2; x3) = x21 +3x22 +5x23 +2x1x2?4x1x3;
(2) f(x1; x2; x3) = x21 +3x23 +2x1x3 +2x2x3;
(3) f(x1; x2; x3) = 2x21 +x22 +4x23 +2x1x2?2x2x3;
3,(1) f = x2
1 +3x
2
2 +5x
2
3 +2x1x2?4x1x3
= (x1 +x2?2x3)2 +2x22 +x23 +4x2x3
= (x1 +x2?2x3)2 +2(x2 +x3)2?x23;
78
>><
>>:
y1 = x1 +x2?2x3;
y2 = p2(x2 +x3);
y3 = x3;
'
8
>><
>>:
x1 = y1?y2=p2+3y3;
x2 = y2=p2?y3;
x3 = y3;
üf1S?f = y21 +y22?y23,
¨MD ?1
C =
0
BB
@
1?1=p2 3
0 1=p2?1
0 0 1
1
CC
A:
(2) f = x2
1 +3x
2
3 +2x1x3 +2x2x3
= (x1 +x3)2 +x23 +2x2x3
= (x1 +x3)2 +(x2 +x3)2?x22;
L?}
(]6
1?)5? Is?91
78
>><
>>:
y1 = x1 +x3;
y2 = x2 +x3;
y3 = x2;
'
8>
><
>>:
x1 = y1?y2 +y3;
x2 = y3;
x3 = y2?y3;
üf1S?f = y21 +y22?y23,
¨MD ?1
C =
0
BB
@
1?1 1
0 0 1
0 1?1
1
CC
A:
(3) f = 2x2
1 +x
2
2 +4x
2
3 +2x1x2?2x2x3
= (x1 +x2?x3)2 +x21 +3x23 +2x1x3
= (x1 +x2?x3)2 +(x1 +x3)2 +2x23;
78
>><
>>:
y1 = x1 +x2?x3;
y2 = x1 +x3;
y3 = p2x3;
'
8
>><
>>:
x1 = y2?y3=p2;
x2 = y1?y2 +p2y3;
x3 = y3=p2;
üf1S?f = y21 +y22 +y23,
¨MD ?1
C =
0
BB
@
0 1?1=p2
1?1 p2
0 0 1=p2
1
CC
A:
31,
!
f(x1; x2; x3) = x21 +x22 +5x23 +2ax1x2?2x1x3 +4x2x3
1??=Q?,pa.
3:?=Q?￥ ?1
A =
0
BB
@
1 a?1
a 1 2
1 2 5
1
CC
A:
???￥ 1Hq,¤
flfl
flfl
fl
1 a
a 1
flfl
flfl
fl > 0;
flfl
flfl
flfl
flfl
1 a?1
a 1 2
1 2 5
flfl
flfl
flfl
flfl
> 0:
'(
1?a2 > 0;
a(5a+4) > 0,)?
4
5 < a < 0:
32,
?/
=Q?￥???:
(1) f =?2x21?6x22?4x23 +2x1x2 +2x1x3;
(2) f = x21 +3x22 +9x23 +19x24?2x1x2 +4x1x3 +2x1x4?6x2x4?12x3x4.
3,(1) f￥ ?1
A =
0
BB
@
2 1 1
1?6 0
1 0?4
1
CC
A;
92??cM
?#=Q?
?
a11 =?2 < 0;
flfl
flfl
fl
2 1
1?6
flfl
flfl
fl = 11 > 0;
flfl
flfl
flfl
flfl
2 1 1
1?6 0
1 0?4
flfl
flfl
flfl
flfl
=?38 < 0;
? ? ?11?f1μ?.
(2) f￥ ?1
A =
0
BB
BB
@
1?1 2 1
1 3 0?3
2 0 9?6
1?3?6 19
1
CC
CC
A;
?
a11 = 1 > 0;
flfl
flfl
fl
1?1
1 3
flfl
flfl
fl = 2 > 0;
flfl
flfl
flfl
flfl
1?1 2
1 3 0
2 0 9
flfl
flfl
flfl
flfl
= 6 > 0; jAj = 24 > 0;
? ? ?11?f1??.
33,￡
ü??A1??￥ sA1Hq
^:i V
I ?U,
PA = UTU,'AD?ê?E
].
￡
ü,( s?)?i V
I ?U,
PA = UTU,?|x 2R,Ox 6= 0,5
Ux 6= 0:
(?TUx = 0,?U V
I,5x = 0.
±.)
???|￥x 6= 0,μ
f(x) = xTAx = xTUTUx = ￡Ux; Ux? = kUxk2 > 0:
V7 ?A1??￥.
(A1?)
!??A1??￥.yA
^??,5i???Q,
PA?,'
QTAQ =? = diag(?1;?2; ￠￠￠ ;?n);
1;?2; ￠￠￠ ;?n1 ?A￥+?′.7A1??￥,
[?i > 0,i = 1; 2; ￠￠￠ ; n.:??
1 = diag(
p
1;
p
2; ￠￠￠ ;
p
n);
5
21 = diag(
p
1;
p
2; ￠￠￠ ;
p
n) diag(
p
1;
p
2; ￠￠￠ ;
p
n) =?:
V7
A = Q?QT = Q?1?1QT =?Q?1￠?Q?1￠T;
:U =?Q?1￠T,5U V
I,7O¤?A = UTU.

课件简介

课件名称：	武汉大学数学与统计学院：线性代数习题全解
课件分类：	数学
课件类型：	试卷习题
文件大小：	499.69KB
下载次数：	33
评论次数：	14
用户评分：	7.1