Question

设A,B为同阶方阵,则（AB）*=?

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*.