x0x1x2x3
x0x7x2 x8x9xa
x1 x2 2–1
1,x7cx12x1dx1ex46x5dx47x48x13x14x46x5d,x47x48x13x2dx46x5d,x47x48x13x3excx46x5d?
(1) f,C?! R
a 7?! jaj
(2) V x22x27x8x70x71,?!e x22x48x49x1dx15x14x2fx1fx20,x46x5d
, V?! V?!
a 7?! (?!a ) =?!a?2(?!a ￠?!e )?!e
(3) f,N?! N
n 7?! n+1
x5,(1)x6ex14,x6ex2d,x55x3exc.
(2)x14,x2d,x3exc.
(2)x14,x6ex2d,x55x3exc.
2,x18fx22xdx54Sx8xdx54S0x15x46x5d,gx22xdx54S0x8xdx54S00x15x46x5d,x53x54:
(1)x8x1fgfx22x14x46x5d,x4afx22x14x46x5d;
(2)x8x1fx22gfx2dx46x5d,x4agx22x2dx46x5d.
x3x4,(1)x2x0x1x15s1;s2 2 S,x8f(s1) = f(s2),x4agf(s1) = gf(s2),x21gfx13x14x46x5d,x21s1 = s2,x43
x25fx13x14x46x5d.
(2) x2x0x1x15s00 2 S00,x21gf x13x2dx46x5d,x21x31x6bs 2 S,x27gf(s) = s00,x43x25s0 = f(s) 2 S0,x27
g(s0) = s00,x21gx13x2dx46x5d.
3,x18fx22xdx54Sx8xdx54S0x15x3excx46x5d,gx22xdx54S0x8xdx54S00x15x3excx46x5d,x4agfx22xdx54Sx8xdx54
S00x15x3excx46x5d,x3f(gf)?1 = f?1g?1.
x3x4,x21x22fx42gx6dx3exc,x23x24f?1g?1x13xdx54S00x8Sx15x48x66x46x5d,x3f
(f?1g?1)(gf) = f?1(g?1g)f = f?11S0f = f?1f = 1S;
(gf)(f?1g?1) = g(ff?1)g?1 = g1S0g?1 = gg?1 = 1S00:
x1 x2 2–2
1,x18:
p =
1 2 3 4 5 6 7
3 5 4 7 6 2 1
; q =
1 2 3 4 5 6 7
2 5 1 7 6 4 3
:
x73pq,p?1qp,x57x4ep;qx11x12x1bx1cx2ax2x4ax15xfx10.
x5,pq =
1 2 3 4 5 6 7
5 6 3 1 2 7 4
; p?1qp =
1 2 3 4 5 6 7
7 5 4 1 3 2 6
,p = (13)(34)(47)(25)(56),
q = (12)(25)(56)(64)(47)(73),(x7xfx37x66x1bx1cx29x55x2cx48).
2,x78x67x1dx1ex59x4ax15xcx3fxe,x57x64x1dx3cx13xdx26:
(1)
1 2 3 4 5 6
3 4 5 6 1 2
;
￠ 1 ￠
(2)
1 4 5 6 3 2 8 7
6 8 5 4 7 2 1 3
;
(3)
2 5 4 3 9 8 6 7 1
6 3 1 2 7 9 8 5 4
;
(4)
1 3 5 ￠￠￠ 2n?1 2 4 6 ￠￠￠ 2n
2 4 6 ￠￠￠ 2n 1 3 5 ￠￠￠ 2n?1
.
x5,(1)xcx3fxex138,xd.
(2)xcx3fxex1320,xd.
(3)xcx3fxex1311,x13.
(4)xcx3fxex13n,x13xdx26x43nx15x13xdx26.
3,x78x67x1dx1ex4bx1ex15xcx3fxe,x57x64x1dx3cx13xdx26:
(1) 5317246; (2) 384576192;
(3) 246813579; (4) 987654321.
x5,(1)xcx3fxex139,x13.
(2)xcx3fxex1318,xd.
(3)xcx3fxex1310,x13.
(4)xcx3fxex1336,xd.
4,x64x1dix68k,x27
(1) 237i864k5x2axdx4bx1e; (2) 469k1i752x2ax13x4bx1e.
x5,(1) i = 1;k = 9.
(2) i = 8;k = 3.
5,x78x67x1dx1ex4bx1ex15xcx3fxe:
(1) 135￠￠￠(2n?1)(2n)(2n?2)￠￠￠642;
(2) (2n+1)(2n)(2n?1)￠￠￠321.
x5,(1) n(n?1).
(2) n(2n+1).
6,xaxbx59x4apx15xcx3fxex22a,x73p?1x15xcx3fxe.
x5,a.
7,xaxbx4bx1ex1x2￠￠￠xnx15xcx3fxex22a,x73xnxn?1￠￠￠x2x1x15xcx3fxe.
x5,x21x22x1x2￠￠￠xnx15xcx3fxe+xnxn?1￠￠￠x2x1x15x4cx3fxe= n(n?1)2,x25xnxn?1￠￠￠x2x1x15xcx3f
xe= x1x2￠￠￠xnx15x4cx3fxe,x23x24xnxn?1￠￠￠x2x1x15xcx3fxe= n(n?1)2?a.
8,x53x54:x2x0x8x55x4dx4e n(n?1)
2 x15x72x2bxek,x40x31x6bxcx3fxex22kx15nx79x4bx1e.
x3x4,x2kx1axex4fx2x50x44.
x51xa,x62k = 1x52,213￠￠￠nx15xcx3fxex221;
x31x1dx22x23x2k?1x2ax2b
k 6 n(n?1)2
,x4x31x6bnx79x4bx1e
i1i2￠￠￠in (1)
x3cxcx3fxex22k?1,x4ax40x31x6bj < k,x27ij < ik (x29x4ax4fx4bx1ex15xcx3fxex22 n(n?1)2 ),x43x25x6bj;kx39x71x40x47
x37x66x65x28x15x52x59j 6 r < r +1 6 k,x27ir < ir+1,x2fx4bx1e
i1￠￠￠ir?1ir+1irir+2￠￠￠in;
￠ 2 ￠
x4ax4fx4bx1ex15xcx3fxe=x4bx1e(1)x15xcx3fxe+1 = k.
x4ex2x50x44x4bx5xbx22x23x2ax2b.
9,x6bx23x47nx79x59x4ax16,x11x12x47x9x46x66xcx3fxex221,2,3x15x59x4a?
x5,x4ex4bx1ex42x59x4ax15x2ax6a,x14x15x7bx53x2x4bx1ex64x1dx65x2cx15x4x4x3e.
x62k = 1x52,x21x0x1xcx3fxex221x15x4bx1ex6dx3ex24x4ex4bx1e123￠￠￠nx3x4ax37x66x65x28x15xex25x50x8,x21xcx3fxe
x221x15x4bx1ex66xex56x3cP1(n) = n?1.
x4ex3cx0x48nx79x4bx1ex6dx3ex24x4en?1x79x4bx1ex54x17xenx25x50x8,x21x62k = 2x52,xcx3fxex222x15x4bx1ex3ex4e
x1dx53x40x29x50x8:
(a) 12￠￠￠n?2 n?1?! 12￠￠￠n?3 n n?2 n?1;
(b) i1i2￠￠￠in?1 (xcx3fxe1)?! i1i2￠￠￠in?2 n in?1;
(c) i1i2￠￠￠in?1 (xcx3fxe2)?! i1i2￠￠￠in?1 n.
x23x24xcx3fxex222x15x4bx1ex66xex22
P2(n) = 1+P1(n?1)+P2(n?1):
x4ex4fx3ex50
P2(n?1) = 1+P1(n?2)+P2(n?2);
::::::::::::::::::::::::::::::::
P2(3) = 1+P1(2)+P2(2);
x23x24P2(n) = (n?2)+(n?1)+￠￠￠+2 = (n?2)(n+1)2,
x62n = 3x52,x6bx6cx3cx79xfx15x55x23,x3ex50
P3(n) = 1+P1(n?1)+P2(n?1)+P3(n?1);
x23x24
P3(n)?P3(n?1) = (n?2)(n+1)2 ;
x4ex4fx3ex50P3(n) = n(n
2?7)
6,
x1 x2 2–3
1,x64x1dx1dx1exdx1ex29x15x3ax4fxfx23x56x15x57x59:
(1) a31a12a23a44; (2) a31a23a14a42a65a56.
x5,(1) +.
(2) +.
2,x1dx1ex28x3ax13x29x22x69x79xdx1ex29x15x3a(x36x37x57x59)?
(1)?a21a34a15a23a52; (2) +a32a15a24a53a41.
x5,(1)x55x13.
(2)x13.
3,x7ex25x6cx79xdx1ex29x16x23x47x56x39x59x3fx36xax21x0a23x15x3a.
x5,?a11a32a23a44,?a31a42a23a14,?a41a12a23a34.
4,x6bnx79xdx1ex29x16,x37x31x2x35x74x79x28xbx58x15xfx10x11x12x2cx7ax2fx30x57x59?
x5,+,(?1)n(n?1)2,
￠ 3 ￠
5,x53x54:
X
i1i2￠￠￠in
sgn(i1i2￠￠￠in) = 0 (n > 2).
x3x4,x21x22x6bx33x10n(n > 2)x79x4bx1ex16,x13xdx4bx1ex28x59x48x5a,x23x24x6bfsgn(i1i2￠￠￠in) j i1;i2;￠￠￠ingx16,
x72x39x59x28x59x48x5a.x21x4f
X
i1i2￠￠￠in
sgn(i1i2￠￠￠in) = 0.
6,x5bx1dx4dx78x67x1dx1exdx1ex29:
(1)
flfl
flfl
flfl
flfl
fl
a 0 0 b
0 c d 0
0 e f 0
g 0 0 h
flfl
flfl
flfl
flfl
fl; (2)
flfl
flfl
flfl
flfl
fl
0 1 0 0
0 0 2 2
3 0 0 3
4 0 0 4
flfl
flfl
flfl
flfl
fl;
(3)
flfl
flfl
flfl
flfl
flfl
fl
a x 0 0 0
0 a x 0 0
0 0 a x 0
0 0 0 a x
x 0 0 0 a
flfl
flfl
flfl
flfl
flfl
fl; (4)
flfl
flfl
flfl
flfl
flfl
fl
0 0 ￠￠￠ 0 1
0 0 ￠￠￠ 2 0
::::::::::::::::::::
0 n?1 ￠￠￠ 0 0
n 0 ￠￠￠ 0 0
flfl
flfl
flfl
flfl
flfl
fl;
(5)
flfl
flfl
flfl
flfl
flfl
fl
0 1 0 ￠￠￠ 0 0
0 0 2 ￠￠￠ 0 0
::::::::::::::::::::::::
0 0 0 ￠￠￠ 0 n?1
n 0 0 ￠￠￠ 0 0
flfl
flfl
flfl
flfl
flfl
fl; (6)
flfl
flfl
flfl
flfl
flfl
fl
1 a a ￠￠￠ a
0 2 a ￠￠￠ a
0 0 3 ￠￠￠ a
::::::::::::::::
0 0 0 ￠￠￠ n
flfl
flfl
flfl
flfl
flfl
fl
.
x5,(1) acfh+bdeg?adeh?bcfg.
(2) 0.
(3) a5 +x5.
(4) (?1)n(n?1)2 n!.
(5) (?1)n?1n!.
(6) n!.
x1 x2 2–4
1,x1ax5cx56xdx3dx4ax76x1dx1ex5dx5ex3dx22x79x34x35x36x5dx5e:
(1)
0
BB
B@
0 4 10 1
4 8 18 7
10 18 40 17
1 7 17 3
1
CC
CA; (2)
0
BB
BB
B@
3 2?1 2 0 1
4 1 0?3 0 2
2?1?2 1 1?3
3 1 3?9?1 6
3?1?5 7 2?7
1
CC
CC
CA;
x5,(1)
0
BB
B@
4 8 18 7
0 4 10 1
0 0 0 0
0 0 0 0
1
CC
CA.
(2)
0
BB
BB
B@
3 2?1 2 0 1
0?1 4?11?1 5
0 0?16 38 5?23
0 0 0 0 0 0
0 0 0 0 0 0
1
CC
CC
CA.
2,x1ax5cx56x1ex3dx4ax76x1dx1ex5dx5ex3dx22x1dx34x35x36x5dx5e:
￠ 4 ￠
(1)
0
BB
B@
2 1?30 5
1 0 4?1
3?2 10?11
1 1?15 8
1
CC
CA; (2)
0
BB
BB
B@
1?1 2 3 4
2 1?1 2 0
1 2 1 1 3
1 5?8?5?12
3?7 8 9 13
1
CC
CC
CA.
x5,(1)
0
BB
B@
2 0 0 0
1 19 0 0
3?35?54 0
1?30 27 0
1
CC
CA.
(2)
0
BB
BB
B@
1 0 0 0 0
2 3 0 0 0
1 1 14 0 0
1 6 0 0 0
3?4?14 0 0
1
CC
CC
CA.
x1 x2 2–5
1,x78x67x1dx1exdx1ex29:
(1)
flfl
flfl
flfl
ab ac ae
bd?cd ed
bf cf?ef
flfl
flfl
flfl; (2)
flfl
flfl
flfl
flfl
fl
3 1 1 1
1 3 1 1
1 1 3 1
1 1 1 3
flfl
flfl
flfl
flfl
fl;
(3)
flfl
flfl
flfl
flfl
fl
1 1 1 1
1 2 4 8
1 3 9 27
1 4 16 64
flfl
flfl
flfl
flfl
fl; (4)
flfl
flfl
flfl
flfl
fl
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
flfl
flfl
flfl
flfl
fl;
(5)
flfl
flfl
flfl
flfl
fl
1+a 1 1 1
1 1?a 1 1
1 1 1+b 1
1 1 1 1?b
flfl
flfl
flfl
flfl
fl; (6)
flfl
flfl
flfl
flfl
flfl
fl
1 1 1 1 1
a1 a a2 a2 a2
a2 a2 a a3 a3
a3 a3 a3 a a4
a4 a4 a4 a4 a
flfl
flfl
flfl
flfl
flfl
fl
.
x5,(1) 4abcdef.
(2) 48.
(3) 12.
(4) 160.
(5) a2b2.
(6) (a?a1)(a?a2)(a?a3)(a?a4).
2,x53x54x1dx1ex56x29:
(1)
flfl
flfl
flfl
sin2 fi cos2 fi cos(2fi)
sin2 fl cos2 fl cos(2fl)
sin2 cos2 cos(2 )
flfl
flfl
flfl = 0;
(2)
flfl
flfl
flfl
b+c c+a a+b
b0 +c0 c0 +a0 a0 +b0
b00 +c00 c00 +a00 a00 +b00
flfl
flfl
flfl = 2
flfl
flfl
flfl
a b c
a0 b0 c0
a00 b00 c00
flfl
flfl
flfl;
￠ 5 ￠
(3)
flfl
flfl
flfl
flfl
fl
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
fl
= 0.
x3x4,(1)x1dx38=
flfl
flfl
flfl
sin2 fi?cos2 fi cos2 fi cos(2fi)
sin2 fl?cos2 fl cos2 fl cos(2fl)
sin2?cos2 cos2 cos(2 )
flfl
flfl
flfl =
flfl
flfl
flfl
cos(2fi) cos2 fi cos(2fi)
cos(2fl) cos2 fl cos(2fl)
cos(2 ) cos2 cos(2 )
flfl
flfl
flfl = 0.
(2)x1dx38= 2
flfl
flfl
flfl
a+b+c c+a a+b
a0 +b0 +c0 c0 +a0 a0 +b0
a00 +b00 +c00 c00 +a00 a00 +b00
flfl
flfl
flfl = 2
flfl
flfl
flfl
a+b+c?b?c
a0 +b0 +c0?b0?c0
a00 +b00 +c00?b00?c00
flfl
flfl
flfl =x1ex38.
(3)x1dx38=
flfl
flfl
flfl
flfl
fl
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
fl
=
flfl
flfl
flfl
flfl
fl
a2 2a+1 2 6
b2 2b+1 2 6
c2 2c+1 2 6
d2 2d+1 2 6
flfl
flfl
flfl
flfl
fl
= 0.
x1 x2 2–6
1,x78x67xdx1ex29
D =
flfl
flfl
flfl
flfl
fl
a 1 2 3
b?1 0 1
c 0 2 3
d 1?1?2
flfl
flfl
flfl
flfl
fl
x15x3dx48x1ex28xbx58x15x51xex24x0x29.
x5,A11 =?1,A21 = 1,A31 = 2,A41 = 2.
2,x73xdx1ex29 fl
flfl
flfl
fl
1?1 2
3 2 1
0 1 4
flfl
flfl
flfl
x15x33x7cxbx58x15x51xex24x0x29.
x5,A11 = 7,A12 =?12,A13 = 3,A21 = 6,A22 = 4,A23 =?1,A31 =?5,A32 = 5,A33 = 5.
3,x78x67x1dx1ex28xdx1ex29:
(1)
flfl
flfl
flfl
flfl
fl
a b c d
a b d c
b a c d
b a d c
flfl
flfl
flfl
flfl
fl; (2)
flfl
flfl
flfl
flfl
flfl
fl
0 1 2?1?4
2 0 1 2 1
1 3?5 1 2
3?3 1 2 1
2 1 0 3 5
flfl
flfl
flfl
flfl
flfl
fl;
(3)
flfl
flfl
flfl
flfl
fl
1 1 1 1
x 0 1 1
x2 1 0 1
x3 1 1 0
flfl
flfl
flfl
flfl
fl; (4)
flfl
flfl
flfl
flfl
fl
x y z t
2 2 2 1
2 2 1 2
2 1 2 2
flfl
flfl
flfl
flfl
fl;
(5)
flfl
flfl
flfl
flfl
fl
a 0 0 b
0 c d 0
0 e f 0
g 0 0 h
flfl
flfl
flfl
flfl
fl
.
x5,(1) 0.
(2)?53.
(3)?x3?x2?x+2.
￠ 6 ￠
(4)?5x+2y +2z +2t.
(5) (ah?bg)(cf?ed).
x1 x2 2–7
1,x1ax5fx60x61x44x4ax2dx1dx1ex74x26x40x41x42:
(1)
8
><
>:
3x1 +2x2 + x3 = 5;
2x1 +3x2 + x3 = 1;
2x1 + x2 +3x3 = 11;
(2)
8>
>>><
>>>
>:
x1 +2x2 +3x3 +4x4 = 0;
x1 + x2 +2x3 +3x4 = 0;
x1 +5x2 + x3 +2x4 = 0;
x1 +5x2 +5x3 +2x4 = 0;
(3)
8
>>>>
<
>>>>
:
x1 +2x2 +3x3?2x4 = 6;
2x1 + x2 +2x3?3x4 = 8;
3x1 +2x2? x3 +2x4 = 4;
2x1 +3x2 +2x3 + x4 = 4;
(4)
8
>>>>
<
>>>>
:
2x1 + x2?5x3 + x4 =?14;
x1?3x2?6x3 =?11;
2x2 + x3 +2x4 = 9;
x1 +4x2 +7x3 +6x4 = 20:
x5,(1) jAj = 12,jB1j = 24,jB2j =?24,jB3j = 36,x1 = 2,x2 =?2,x3 = 3.
(2) jAj =?20,jB1j = jB2j = jB3j = jB4j = 0,x1 = x2 = x3 = x4 = 0.
(3) jAj = 12,jB1j = 12,jB2j = 24,jB3j =?12,jB4j =?24,x1 = 1,x2 = 2,x3 =?1,x4 =?2.
(4) jAj =?9,jB1j =?9,jB2j = 18,jB3j =?27,jB4j =?9,x1 = 1,x2 =?2,x3 = 3,x4 = 1.
2,x73x48x66x62x48x9x3ax29f(x),x27f(1) = 1,f(?1) = 9,f(2) = 3.
x5,f(x) = 2x2?4x+3.
3,x53x54x3x48x74x26x40x41x42 8
>>>>
<
>>>>
:
x1 + x2 +￠￠￠+ xn = 0;
2x1 +22x2 +￠￠￠+2nxn = 0;
￠￠￠￠￠￠￠￠￠￠￠￠
nx1 +n2x2 +￠￠￠+nnxn = 0
x63x47x6fx2d.
x3x4,x21x22
jAj =
flfl
flfl
flfl
flfl
fl
1 1 ￠￠￠ 1
2 22 ￠￠￠ 2n.
..,..,..,..
n n2 ￠￠￠ nn
flfl
flfl
flfl
flfl
fl
= 1!2!￠￠￠n! 6= 0;
x3dx3ex5ex237.2,x4bx40x41x7bx47x6fx2d.
x1 x2 2–8
1,x76x1dx1exdx1ex29x5bx60x63x60x64x1dx5x65x66,x24x73x1dx1exdx1ex29x15x4:
(1)
flfl
flfl
flfl
flfl
flfl
fl
5 6 0 0 0
1 5 6 0 0
0 1 5 6 0
0 0 1 5 6
0 0 0 1 5
flfl
flfl
flfl
flfl
flfl
fl; (2)
flfl
flfl
flfl
flfl
fl
a 0 b 0
0 c 0 d
y 0 x 0
0 w 0 z
flfl
flfl
flfl
flfl
fl;
￠ 7 ￠
(3)
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 1 1 0 0 0
2 3 4 0 0 0
3 10 16 0 0 0
1 1 0 1 1 1
2 4 1 1 2 3
3 16 1 1 4 9
flfl
flfl
flfl
flfl
flfl
flfl
fl;
(4)
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
a 0 0,::::::::::::,0 0 b
0 a 0,::::::::::::,0 b 0
0 0 a,::::::::::::,b 0 0.
...
...
...
...
.
...
...
...
...
..
...
...
...
...
..
...,..,..
...
...
...
..
...
...
...
...
..
...
...
...
...
..
a b
b a
...,..
0 0 b,::::::::::::,a 0 0
0 b 0,::::::::::::,0 a 0
b 0 0,::::::::::::,0 0 a
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
flfl
2n
.
x5,(1)x5bx3d1,2x37xdx65x66:
x4bx29=
flfl
flfl5 6
1 5
flfl
flfl￠
flfl
flfl
flfl
5 6 0
1 5 6
0 1 5
flfl
flfl
flfl+(?1)7
flfl
flfl5 0
1 6
flfl
flfl￠
flfl
flfl
flfl
1 6 0
0 5 6
0 1 5
flfl
flfl
flfl+(?1)8
flfl
flfl6 0
5 6
flfl
flfl￠
flfl
flfl
flfl
0 6 0
0 5 6
0 1 5
flfl
flfl
flfl = 665:
(2) (ax?by)(cz?dw).
(3)x5bx3d1,2,3xdx65x66:
x4bx29=
flfl
flfl
flfl
1 1 1
2 3 4
3 10 16
flfl
flfl
flfl￠
flfl
flfl
flfl
1 1 1
1 2 3
1 4 9
flfl
flfl
flfl =?2:
(4)x47x48x5bx16x71x37xdx65x66:
x4bx29= (a2?b2)D2(n?1) = ￠￠￠ = (a2?b2)n:
2,x78x67x1dx1exdx1ex29x15x4:
(1)
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 a1 0 ￠￠￠ 0 0
1 1?a1 a2 ￠￠￠ 0 0
0?1 1?a2 ￠￠￠ 0 0
::::::::::::::::::::::::::::::::::::::::::::
0 0 0 ￠￠￠ 1?an?1 an
0 0 0 ￠￠￠?1 1?an
flfl
flfl
flfl
flfl
flfl
flfl
fl;
(2)
flfl
flfl
flfl
flfl
flfl
flfl
fl
n?1 0 0 ￠￠￠ 0 0
n?1 x?1 0 ￠￠￠ 0 0
n?2 0 x?1 ￠￠￠ 0 0
::::::::::::::::::::::::::::::::::
2 0 0 0 ￠￠￠ x?1
1 0 0 0 ￠￠￠ 0 x
flfl
flfl
flfl
flfl
flfl
flfl
fl;
(3)
flfl
flfl
flfl
flfl
flfl
fl
x a a ￠￠￠ a a
a x a ￠￠￠ a a
a?a x ￠￠￠ a a
:::::::::::::::::::::::::::
a?a?a ￠￠￠?a x
flfl
flfl
flfl
flfl
flfl
fl;
￠ 8 ￠
(4)
flfl
flfl
flfl
flfl
flfl
flfl
fl
x y y ￠￠￠ y y
z x y ￠￠￠ y y
z z x ￠￠￠ y y
::::::::::::::::::::
z z z ￠￠￠ x y
z z z ￠￠￠ z x
flfl
flfl
flfl
flfl
flfl
flfl
fl;
(5)
flfl
flfl
flfl
flfl
flfl
flfl
fl
a+b b 0 ￠￠￠ 0 0
a a+b b ￠￠￠ 0 0
0 a a+b ￠￠￠ 0 0
::::::::::::::::::::::::::::::::::::::
0 0 0 ￠￠￠ a+b b
0 0 0 ￠￠￠ a a+b
flfl
flfl
flfl
flfl
flfl
flfl
fl
,(a 6= b);
(6)
flfl
flfl
flfl
flfl
flfl
flfl
fl
1 2 3 ￠￠￠ n?1 n
2 3 4 ￠￠￠ n 1
3 4 5 ￠￠￠ 1 2
::::::::::::::::::::::::::::::::
n?1 n 1 ￠￠￠ n?3 n?2
n 1 2 ￠￠￠ n?2 n?1
flfl
flfl
flfl
flfl
flfl
flfl
fl
.
x5,(1)x28xdx17x8x3d1xd,x50D = (?1)n+2 ￠(?1)n = 1.
(2)x67x3d1xdx74,x28xdxfx24xx17x8x1dx48xd:
D = (?1)n+1(1+2xcdots+nxn?1)￠(?1)n?1 = 1+2x+￠￠￠+nxn?1:
(3) Dn =
flfl
flfl
flfl
flfl
flfl
fl
a a a ￠￠￠ a a
a x a ￠￠￠ a a
a?a x ￠￠￠ a a
:::::::::::::::::::::::::::
a?a?a ￠￠￠?a x
flfl
flfl
flfl
flfl
flfl
fl
+
flfl
flfl
flfl
flfl
flfl
fl
x?a a a ￠￠￠ a a
0 x a ￠￠￠ a a
0?a x ￠￠￠ a a
:::::::::::::::::::::::::::::
0?a?a ￠￠￠?a x
flfl
flfl
flfl
flfl
flfl
fl
=
flfl
flfl
flfl
flfl
flfl
fl
a a a ￠￠￠ a a
0 x a ￠￠￠ a a
0 0 x ￠￠￠ a a
::::::::::::::::::::
0 0 0 ￠￠￠ 0 x
flfl
flfl
flfl
flfl
flfl
fl
+(x?a)Dn?1,
x23x24
Dn = a(x+a)n?1 +(x?a)Dn?1;
x51,
Dn = DTn = a(x?a)n?1 +(x+a)Dn?1;
x6x7Dn?1,x50
Dn = 12 [(x+a)n +(x?a)n]:
(4)x6bx6cx3cx79x61,x3ex50
Dn = z(x?y)n?1 +(x?z)Dn?1:
x51
Dn = DTn = y(x?z)n?1 +(x?y)Dn?1:
x62y 6= zx52,x4ex79x37x29x6x7Dn?1,x50
Dn = y(x?z)
n?z(x?y)n
y?z,
￠ 9 ￠
x62y = zx52,x4ex68x5ex66x29Dn = y(x?y)n?1 +(x?y)Dn?1,x50Dn = (x+(n?1)y)(x?y)n?1.
(5)x49?0 = 1,?1 = a+b = a2?b2a?b,:::,?n = an+1?bn+1a?b,x4a
(a+b)?k?ab?k?1 =?k+1:
Dn =
flfl
flfl
flfl
flfl
flfl
fl
1 b?0 0 ￠￠￠ 0 0
a a+b b ￠￠￠ 0 0
::::::::::::::::::::::::::::::::
0 0 0 ￠￠￠ a+b b
0 0 0 ￠￠￠ a a+b
flfl
flfl
flfl
flfl
flfl
fl
n
=
flfl
flfl
flfl
flfl
flfl
fl
1 b 0 ￠￠￠ 0 0
a a+b? ab?0?
1
b ￠￠￠ 0 0
:::::::::::::::::::::::::::::::::::::::::
0 0 0 ￠￠￠ a+b b
0 0 0 ￠￠￠ a a+b
flfl
flfl
flfl
flfl
flfl
fl
n
=
flfl
flfl
flfl
flfl
flfl
fl
(a+b)?1?ab?0 b?1 0 ￠￠￠ 0 0
a a+b b ￠￠￠ 0 0
:::::::::::::::::::::::::::::::::::::::::::::::
0 0 0 ￠￠￠ a+b b
0 0 0 ￠￠￠ a a+b
flfl
flfl
flfl
flfl
flfl
fl
n?1
=
flfl
flfl
flfl
flfl
flfl
fl
2 b?1 0 ￠￠￠ 0 0
a a+b b ￠￠￠ 0 0
::::::::::::::::::::::::::::::::
0 0 0 ￠￠￠ a+b b
0 0 0 ￠￠￠ a a+b
flfl
flfl
flfl
flfl
flfl
fl
n?1
= ￠￠￠ =?n = a
n+1?bn+1
a?b,
(6)
Dn =
flfl
flfl
flfl
flfl
flfl
flfl
fl
n(n+1)
2 2 3 ￠￠￠ n?1 nn(n+1)
2 3 4 ￠￠￠ n 1::::::::::::::::::::::::::::::::::::
n(n+1)
2 n 1 ￠￠￠ n?3 n?2n(n+1)
2 1 2 ￠￠￠ n?2 n?1
flfl
flfl
flfl
flfl
flfl
flfl
fl
= n(n+1)2
flfl
flfl
flfl
flfl
flfl
fl
1 2 3 ￠￠￠ n?1 n
1 3 4 ￠￠￠ n 1
::::::::::::::::::::::::::::
1 n 1 ￠￠￠ n?3 n?2
1 1 2 ￠￠￠ n?2 n?1
flfl
flfl
flfl
flfl
flfl
fl
= n(n+1)2
flfl
flfl
flfl
flfl
fl
1 2 3 ￠￠￠ n?1 n
0 1 1 ￠￠￠ 1 1?n
::::::::::::::::::::::::::::::::
0 1?n 1 ￠￠￠ 1 1
flfl
flfl
flfl
flfl
fl
= n(n+1)2
flfl
flfl
flfl
flfl
fl
1 1 ￠￠￠ 1 1?n
1 1 ￠￠￠ 1?n 1
::::::::::::::::::::::::::::
1?n 1 ￠￠￠ 1 1
flfl
flfl
flfl
flfl
fl
n?1
= (?1) n(n+1)2
flfl
flfl
flfl
flfl
fl
1 1 ￠￠￠ 1 1?n
1 1 ￠￠￠ 1?n 1
::::::::::::::::::::::::
1 1 ￠￠￠ 1 1
flfl
flfl
flfl
flfl
fl
n?1
= (?1)n(n?1)2 n
n?1(n+1)
2,
￠ 10 ￠