If A, B and A+B are idempotent matrices, then prove that AB+BA=0?

1 Answer
May 30, 2018

See below

Explanation:

By idempotency of A+B:

(A+B)^2 = A+B = square

Distributing:

(A+B)^2 = (A+B)(A+B)

= A^2 + AB + BA + B^2

By idempotency of A, B:

= A + AB + BA + B = triangle

triangle = square implies AB + BA = 0