设A,B为N阶方阵,且A=1/2(B+E),证明A^2=A,当且仅当B^2=E

Question

设A,B为N阶方阵,且A=1/2(B+E),证明A^2=A,当且仅当B^2=E

Answer 1

A^2=[1/2(B+E)]^2
=1/4(B^2+2B+E)=A=1/2(B+E)
化简得
B^2=E