第一章 行列式 一.?填空题? 1.?四阶行列式中带有负号且包含?a12?和?a21?的项为______.?解.?a12a21a3a4?中行标的排列为?1234,?逆序为?0??列标排列为?2134,?逆序为1.?该项符号为“－”,?所 以答案为?a 12a21a3a4.?2.?排列?i1i2…in?可经______次对换后变为排列?inin－1…i2i1.?解.?排列?i 1i2…in?可经过?1?+?2?+?…?+?(n－1)?=?n(n－1)/2?次对换后变成排列?inin－1…i2i1.?3.?在五阶行列式中? 35?24?41?53?12?)?23145?(?)?15423?(?)?1(? a?a?a?a?a t t + -? =______?35?24?41?53?12? a?a?a?a?a? .? 解.?15423?的逆序为?5,?23145?的逆序为?2,?所以该项的符号为“－”.?4.?在函数? x?x?x?x?x?x?f? 2?1? 1?1?2?)?( - - - = 中,?x?3?的系数是______.?解.?x? 3?的系数只要考察? 2?3? 4?2?2?2? x?x?x?x?x?x + - = - -? .?所以?x?3?前的系数为?2.? 5.?设?a,?b?为实数,?则当?a?=?______,?且?b?=?______时,? 0?1?0?1? 0?0 = - - -? a?b?b?a? .? 解.? 0?)?(?1?1?0?1? 0?0? 2?2 = + - = - - = - - -? b?a?a?b?b?a?a?b?b?a? .?所以?a=?b?=?0.? 6.?在?n?阶行列式?D=?|aij|中,?当?i?<?j?时?aij?=?0?(i,?j?=1,?2,?…,?n),?则?D?=?______.? 解.? n? n?n? a?a?a?aaa?a?a? L L O M L L? 2?1? 2?1? 2?21?1? 0?0?0?0 =? 7.?设A为3×3矩阵,?|A|?=－2,?把A按行分块为 ú ú ú ? ù ê ê ê ? é =? 3?2?1?A?A?A?A? ,?其中Aj?(j?=?1,?2,?3)是A的第j行,?则行列 式 = -?1?2?1?3?3?2?A?A?A?A ______.? 解. = -?1?2?1?3?3?2?A?A?A?A 6?|?|?3?3?2?3? 3?2?1?1?2? 1?3 = - = - = -? A?A?A?A?A?A?A?A? .? 二．计算证明题? 1.?设? 4?3?2?2? 3?2?1?1? 4?3?1?1? 3?1?5?1?|?| - =?A 计算?A41?+?A42?+?A43?+?A4?=??,?其中?A4j(j=?1,?2,?3,?4)是|A|中元素?a4j?的代数余子式.? 解.?A41?+?A42?+?A43?+?A4? 1?1?1?1? 3?2?1?1? 4?3?1?1? 3?1?5?1 - =? 2?1?0?3?2?0?2?0?6?)?1?(?0?0?0?1? 2?1?0?1? 3?2?0?1? 2?0?6?1? 1?4 - - = - = +? =? 6?2?1?0?0? 3?2?0? 2?0?6?1 = - -? 2.?计算元素为?aij?=?|i－j|的?n?阶行列式.? 解.? 1?1?1? 1?1?1? 1?1?0?0?2?1? 2?0?1? 1?1?0?|?| - - - - - - - - = L O M L L L O M L L? n?n?n? n?n?A? 每行减前一行 由最后一行起，? )?1?(?2?)?1?(?1?0?0?0? 2?0? 1?2?0? 1?1? 2?1 - - = - - - - - - - -? n?n?n?n?n? n?n?L M M L L O O M L L L L 列 每列加第? 3.?计算?n阶行列式? n?x?x?x? n?x?x?x? n?x?x?x?D? n?n?n? n + + + + + + + + + = L L L L L L L? 2?1? 2?1? 2?1? 2?2?2? 1?1?1? (n 3?2).? 解.?当? 2 >?n? n?x?x?x? n?x?x?x? n?x?x?x?D? n?n?n? n + + + + + + = L L L L L L L? 2?2? 2? 2?2?2? 1?1?1? +? n?x?x? n?x?x? n?x?x? n?n + + + + + + L L L L L L L?2?1? 2?1? 2?1? 2?2? 1?1? =? n?x?x?x?x? n?x?x?x?x? n?x?x?x?x? n?n?n?n + + + + + + L M M M M M L L?3?3?3? 2?2?2?2? 1?1?1?1? +? n?x?x?x? n?x?x?x? n?x?x?x? n?n?n + + + + + + L M M M M M L L?3?2? 3?2? 3?2? 2?2?2? 1?1?1? +? n?x?x?x? n?x?x?x? n?x?x?x? n?n?n + + + + + + L M M M M M L L?3?1? 3?1? 3?1? 2?2?2? 1?1?1? +? n?x?x? n?x?x? n?x?x? n?n + + + + + + L M M M M M L L?3?2?1? 3?2?1? 3?2?1? 2?2? 1?1? =－? n?x?x?x? n?x?x?x? n?x?x?x? n?n?n + + + + + + L M M M M M L L?3?1? 3?1? 3?1? 2?2?2? 1?1?1? =－? n?x?x?x? n?x?x?x? n?x?x?x? n?n?n + + + L M M M M M L L?1?1?1? 2?2?2? 1?1?1? －? n?x?x? n?x?x? n?x?x? n?n + + + L M M M M M L L?3?1? 3?1? 3?1? 2?2? 1?1? =?0 当? 2 =?n? 2?1?2?2? 1?1? 2?1? 2?1? x?x?x?x? x?x - = + + + +?4.证明:奇数阶反对称矩阵的行列式为零.? 证明:? |?|?|?|?)?1?(?|?|?|?|?|?|?,? A?A?A?A?A?AA? n?T?T - = - = - = = - =? (n?为奇数).?所以|A|?=?0.?5.试证:?如果n次多项式? n? n?x?Cx?C?C?x?f? L + + =? 1?0?)?(? 对n?+?1个不同的x值都是零,?则此多项式恒等 于零.?(提示:?用范德蒙行列式证明)?证明:?假设多项式的?n+?1?个不同的零点为?x 0,?x1,?…,xn.?将它们代入多项式,?得关于?Ci?方程组?0?0?0?1?0 = + +? n?n?x?Cx?CC? L? 0?1?1?1?0 = + +? n?n?x?Cx?CC? L …………? 0?1?0 = + +? n?n?n?n? x?Cx?CC? L 系数行列式为?x0,?x1,?…,xn?的范德蒙行列式,?不为?0.?所以? 0?1?0 = = = =? n?CCC? L? 6.?设? ).?(?'?,?6?2?0? 3?2?1?)?(? 2?3?2? x?F?x?x?x?x?x?x?x?F? 求 = 解.? x?x?x?x?x?x?x?F? 6?2?0? 3?2?1?)?(? 2?3?2 =? =? x?x?x?x?x?x? 3?1?0?3?2?1?1?2? 2?2?=? x?x?x?x?x?x? 3?1?0?2?0?1?2? 2?2?=? x?x?x?x?x? 3?1?0?2?1?0?1?2? 2?2? =? 3?2?2? 2?0?0?2?1?0?1?2? x?x?x?x?x?x =? 2?6?)?(?'? x?x?F =