Question

设AB为方阵 分块矩阵C= A 0 ,则C*=?

Answer 1

假设A为n阶方阵,B为m阶方阵

设C*= [ D H ]

[ F G ]

|C|=|A||B|

根据伴随矩阵的性质有：

CC*=C*C=|C|E=|A||B|E

CC*=[ A 0 ] [ D H ] = [ AD AH ] = [ |AB|En 0 ]

[ 0 B ] [ F G ] [ BF BG ] [ 0 |AB|Em]

比较各项得:AD=|AB|E,AH=0,BF=0,BG=|AB|E

于是H=0,F=0

从AD=|AB|E可得到A*AD=A*|AB|E

|A|D=A*|AB|E,

D=|B|A*

同理BG=|AB|E得到B*BG=B*|AB|E

G=|A|B*

故C*=[ |B|A* 0 ]

[ 0 |A|B* ]