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

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Answer 1 · 2018-05-30T14:25:58

See below

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#