The matrix A^2+4 A-5 I, where I is identity matrix and A= [ … ], equals :

$$\begin{aligned} & \mathrm{A}^2+4 \mathrm{~A}-5 \mathrm{I}=\mathrm{A} \times \mathrm{A}+4 \mathrm{~A}-5 \mathrm{I} \\ = & {\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right] \times\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]+4\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]-5\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] } \\ & =\left[\begin{array}{cc}9 & -4 \\ -8 & 17\end{array}\right]+\left[\begin{array}{cc}4 & 8 \\ 16 & -12\end{array}\right]-\left[\begin{array}{cc}5 & 0 \\ 0 & 5\end{array}\right] \\ & =\left[\begin{array}{cc}9+4-5 & -4+8-0 \\ -8+16-0 & 17-12-5\end{array}\right]=\left[\begin{array}{ll}8 & 4 \\ 8 & 0\end{array}\right] \\ & =4\left[\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}\right]\end{aligned}$$=​A2+4 A−5I=A×A+4 A−5I[14​2−3​]×[14​2−3​]+4[14​2−3​]−5[10​01​]=[9−8​−417​]+[416​8−12​]−[50​05​]=[9+4−5−8+16−0​−4+8−017−12−5​]=[88​40​]=4[22​10​]​