Question

If , find A² − 5A − 14I.

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{cc}3& -5\\ -4& 2\end{array}\right] \\ \mathrm{Now}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{cc}3& -5\\ -4& 2\end{array}\right]\left[\begin{array}{cc}3& -5\\ -4& 2\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}9+20& -15-10\\ -12-8& 20+4\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}29& -25\\ -20& 24\end{array}\right] \\ {A}^2-5A-14I \\ \Rightarrow A^2-5A-14I=\left[\begin{array}{cc}29& -25\\ -20& 24\end{array}\right]-5\left[\begin{array}{cc}3& -5\\ -4& 2\end{array}\right]-14\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow A^2-5A-14I=\left[\begin{array}{cc}29& -25\\ -20& 24\end{array}\right]-\left[\begin{array}{cc}15& -25\\ -20& 10\end{array}\right]-\left[\begin{array}{cc}14& 0\\ 0& 14\end{array}\right] \\ \Rightarrow A^2-5A-14I=\left[\begin{array}{cc}29-15-14& -25+25+0\\ -20+20+0& 24-10-14\end{array}\right] \\ \Rightarrow A^2-5A-14I=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \end{gathered}$$