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#