A＝½(B＋E)证明A²＝A的充分必要条件是B²＝E矩阵

Question

A＝½(B＋E) 证明A²＝A的充分必要条件是B²＝E 矩阵

Answer 1

由A²＝A得,1/4(B＋E)^2=1/4(B^2+2B+E)=½(B＋E),因此,1/4B^2=1/4E,所以B^2=E.
由于B^2=E,所以A^2=1/4(B^2+2B+E)=1/4(2B+2E)=1/2(B+E)=A.