第1章行列式 ·23 0 a an 1-1 8 -a 0 a2w-1 (8)D. (n为奇数) 41-1 一a2w-1 一as-1 0 da-Ia - 一aw -di 一a+t 0 21 0. 00 1 2 1. 0 0 (9)D. 2 . 0 0 . 0.: 0 0 . 2 0 1 2 . n-2 n-1 1 0 1 n-3 n-2 (10)D. 2 1 0 n-4 n-3 m-1 n-2 n-3 0 :000 ax a . 0 (11)D, ax"-I ar 13 解(1) 12 5292 2545 13121 2545 5 一1 =号× 1312 2 4 5 -5 -15 -9 -5
24 线性代数学习指导 0 -5 1 420 -4 一4 一1 -5 一1 2 2536 43 0 0 4 2 31 -2 0 -2 03 -2 一5 41 -40 0 10 510 -2 3 =2 =90 0 5 -7-5 3111 6111 1111 1111 (3) 1 3 11 131 6 0200 1 1 3 1 13 =48 020 11 6 1 13 11 3/ 002 a 0 b 0 d c0 d o c d (4) 0 =a x 0+by 00 x 0 0e0 u -ax c -by (ax-by)(cu-dw) w u -1 -11x-1 -1 -1 0 (5) x+1 n-n 1 x-1 1 -1 0 x+1 1 -1 x 0 0 -11 0 1-1 x 0-x 0 -x x00-x 11 -1 x-I 0 x-1 -r x 0 -x =-x2(-x-x2+x)=x
第1章行列式 ·25· a 0 0 . 0 0 a 0 0 0 0 . 0 (6)D.= 0 按最后一行展开 0 0 0 0 0 . 0 a 0 00.01 0 0.0 0 00.0 0 0 0 0 0 (-1)4 0 a0. 0 0 +(-1)2·a 0 0 a 0 0 0 00.a 0 0 0 0 0 a 0 0 .0 0 0 a 0 0 0 =(-1)+H·(-1)” 0: 0 a 0 0 十a"=d-d2=d2(a2-1): 0 0 0 0 a (7)解法一 1 1 1 1+a 1 D.= 1 1 1+a 1 1 1 1 1 1+a. a 1 1 0 1十a 1 + 0 1 1+a . 1 /0 1 1 .11十4. 1 11 .11 0 ag0. 0 0 00a4.0 0+aD.- 000 .0 =azasan+aD
·26· 线性代数学习指导 递推公式 Dn=a2aaan十a1D.-1=a2agan+a(aaa1.an十azDn-2) =a2agan十aasaa.十a1a2(a"an十aD,-z) =a2a.an十dzda.十.十a1a2an-8an-zD2 而D2= 1+aw-11 =an十aw-1十aa4-1am,所以 11+an n.-aara.+会》月 解法二先化简再拆项， |1+a111.1 1 -aa20.0 0 r-n -a 0 as 0 0 D,=2,3.m::氵 -a100. - 0 -a100 . 0 1111.1 1/ 11.1 1 0ag0.0 0 -aj 0 . 0 0 00 a3 .0 0 0 0 0 = 000.a-1 0 -a1 0 0 . an-1 000. 0 -a1 0 0 0 1 1 . 1 1 as-1 -1 0 0 0 =a2a3am十a1a2.an -1 0 0 0 -1 0 0 1 0 -1 0 0 1
第1章行列式 ·27· 1+2. a az a dn-l dn 0 10.0 0 =a2a3a。十a1a2an 0 01 0 0 0 00.1 0 0 00. 0 =aaa+aaa+空）=aea+空》 解法三构造n十1阶行列式 1 1 1 1. 1 11 01+a 1 1. 1 0 1 H.= 11+a21 0 1 11.1+a1 1 0 1 11. 1 1+a, 显然 111 1. 1 1 -1a400. 0 0 ,一1 D.=H-10=2.3.m+D -10ag 0 0 -1000. 0:0 -1000 . 0 0 1+是11 0 a0. 0 (行=2,3.，m+1D 0 0a2. 0 0 00.a. =aaa+》 (8)D,=(一1)"D=(一1)D.,而n为奇数，则D,=一D.,所以D,=0