{\displaystyle {\begin{aligned}\\&A^{2}-(a+d)A+(ad-bc)I_{2}\\[1pt]\\&={\begin{pmatrix}a^{2}+bc&ab+bd\\ac+cd&bc+d^{2}\end{pmatrix}}-{\begin{pmatrix}a(a+d)&b(a+d)\\c(a+d)&d(a+d)\end{pmatrix}}+(ad-bc)I_{2}\\[1pt]\\&={\begin{pmatrix}bc-ad&0\\0&bc-ad\\\end{pmatrix}}+(ad-bc)I_{2}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.\end{aligned}}}
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