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

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Answer 1 · 2018-05-30T14:25:58

See below

By idempotency of $A + B$ $A+B$ :

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

Distributing:

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

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

By idempotency of $A, B$ :

$= A + AB + BA + B = triangle$

$triangle = square implies AB + BA = 0$

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