MT$2\mathrm{.}1(37$ Points$)$ Consider a square $\backslash ($ n $\backslash$times n $\backslash )$ matrix $\backslash (\backslash$mathbf$\{$P$\}\backslash )$ having all of the following properties:

$-$ Each entry of $\backslash (\backslash$mathbf$\{$P$\}\backslash )$ is either $0$ or $1.$

$-$ Each column of $\backslash (\backslash$mathbf$\{$P$\}\backslash )$ contains exactly one $1.$

$-$ Each row of $\backslash (\backslash$mathbf$\{$P$\}\backslash )$ contains exactly one $1.$

$($a$)(10$ Points$)$ Show that $\backslash (\backslash$mathbf$\{$P$\}^\{\backslash$top$\}\backslash$mathbf$\{$P$\}=\backslash$mathbf$\{$I$\}\backslash ).$

We'll accept any logically$-$sound explanation. You need not resort to elaborate mathematical derivations. It's possible to explain this result in two to three sentences.

Even if you have trouble showing this result, you may use it in the subsequent parts of the problem, if you find it useful.