大一数学题.LetAbeasquarematrixsatisfyingA^2=AandletBbeanymatrixofthesamesize.Showthat(AB-ABA)^2=0A是一个满足A^2=A的方矩阵,B是同样大小的任意矩阵.证明(AB-ABA)^2=0.应该是这个意思,请指点思路,

Question

大一数学题.
Let A be a square matrix satisfying A^2=A and let B be any matrix of the same size.Show that (AB-ABA)^2=0
A是一个满足A^2=A的方矩阵,B是同样大小的任意矩阵.证明(AB-ABA)^2=0.应该是这个意思,请指点思路,

Answer 1

(AB-ABA)^2
=ABAB-AB*ABA-ABA*AB+ABA*ABA
=ABAB-ABABA-AB(A^2)B+AB(A^2)BA
=ABAB-ABABA-ABAB+ABABA
=0