设A,B为n阶矩阵,且满足A^2=A,B^2=B,(A+B)^2=(A+B),证明:AB=0.

问题描述:

设A,B为n阶矩阵,且满足A^2=A,B^2=B,(A+B)^2=(A+B),证明:AB=0.

A^2=A,B^2=B,
(A+B)^2=(A+B)==>AB+BA=0==>0=A^2B+ABA=AB+ABA,0=ABA+BA^2=ABA+BA
===>ABA=-AB=-BA
==>AB=BA