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## Homework Statement

A square matrix P is called an idempotent if P^2 = P,

Show that if P is an idempotent, so is Q = (P + AP - PAP) for any square matrix A (of the

same size as P).

## Homework Equations

## The Attempt at a Solution

basically I factor,

Q = (I + (I - P) A) P

then I square it and try to get back to the original,... and end up with

Q^2 = (I + (I - P - P + P^2) A^2) P^2

= (I + (I -2P + P) A^2) P

= (I + (I - P) A^2) P

= (I + AA - PAA) P

= P + AAP - PAAP

how do i get rid of the AA (A^2) ..?

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