设A,B为同阶方阵,则（AB）*=?

Question

如题所述

Answer 1

知识点:对任一n阶方阵A,总有 AA*=A*A=|A|E
当A,B可逆时
|A||B|B*A*
= |AB|E(B*A*)
= (AB)*AB(B*A*)
= (AB)*A(BB*)A*
= (AB)*A|B|EA*
= |A||B|(AB)*.
∵ |A|≠0,|B|≠0,
∴ (AB)*=B*A*.

Answer 2

简单分析一下，答案如图所示