Show that $(I − P)^2 = I − P$ if $P=P^2$

Question

Let $P $ be an $n \times n$ matrix and $I$ be the $n \times n$ identity matrix. Show that $$ (I − P)^2 = I − P $$ is valid if $P = P^2$.

I did the following.

$$(I - P)^2 = I^2 - IP - PI + P^2 = I - P$$

where $I^2 = I $ because it is the identity matrix. Is this enough to show or did I miss something?

Answer 1

I can only speculate about what "sth" means (I thought it indicated the $s$th element of some sequence, at first, for some natural number $s$) but I'd say you are close!

I'd be a bit more explicit at this stage.

\begin{eqnarray}(I-P)^2 &=& I^2-IP-PI+P^2\\ &=& I-P-P+P^2\\ &=& I-P-P+P\\ &=& I-P\end{eqnarray}

Can you justify each equality?