Question

设矩阵A=（1,1,-1；-1,1,1；1,-1,1）的伴随矩阵为A*,矩阵B满足A*B=A^-1+2B,求B

Answer 1

由 A*B=A^-1+2B
左乘A得 |A|B = E + 2AB
由 |A| = 4
得 (4E-2A)B = E.
(4E-2A,E) =
2 -2 2 1 0 0
2 2 -2 0 1 0
-2 2 2 0 0 1
r2-r1,r3+r1
2 -2 2 1 0 0
0 4 -4 -1 1 0
0 0 4 1 0 1
r1*(1/2),r2*(1/4),r3*(1/4)
1 -1 1 1/2 0 0
0 1 -1 -1/4 1/4 0
0 0 1 1/4 0 1/4
r1+r2,r2+r3
1 0 0 1/4 1/4 0
0 1 0 0 1/4 1/4
0 0 1 1/4 0 1/4
B = (4E-2A)^-1 =
1/4 1/4 0
0 1/4 1/4
1/4 0 1/4