Es ist

${\displaystyle {}{\begin{aligned}P_{M}(M)&={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\circ {\begin{pmatrix}a&b\\c&d\end{pmatrix}}-(a+d){\begin{pmatrix}a&b\\c&d\end{pmatrix}}+(ad-bc){\begin{pmatrix}1&0\\0&1\end{pmatrix}}\\&={\begin{pmatrix}a^{2}+bc&ab+bd\\ac+dc&cb+d^{2}\end{pmatrix}}-{\begin{pmatrix}a(a+d)&b(a+d)\\c(a+d)&d(a+d)\end{pmatrix}}+{\begin{pmatrix}ad-bc&0\\0&ad-bc\end{pmatrix}}\\&={\begin{pmatrix}a^{2}+bc-a(a+d)+ad-bc&ab+bd-b(a+d)\\ac+dc-c(a+d)&cb+d^{2}-d(a+d)+ad-bc\end{pmatrix}}\\&={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.\end{aligned}}}$