设矩阵A=（1,0,0；0,-2,0；0,0,1）的伴随矩阵为A*矩阵B满足A*BA=2BA-8I,求B.
问题描述：
设矩阵A=（1,0,0；0,-2,0；0,0,1）的伴随矩阵为A*矩阵B满足A*BA=2BA-8I,求B.
答
易知 |A|=-2,A可逆.由 A*BA=2BA-8I,左乘A,右乘A^-1,得AA*BAA^-1=2ABAA^-1-8AA^-1所以 |A|B = 2AB - 8I所以 (A+I)B = 4I所以 B = 4(A+I)^-1 = 4*1/2 0 00 -1 00 0 1/2=2 0 00 -4 00 0 2