Question

If $A=\left[\begin{array}{cc}1& -3\\ -4& 3\end{array}\right]$, find A² − 3A − 7I.

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{cc}1& -3\\ -4& 3\end{array}\right] \\ \mathrm{Now}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{cc}1& -3\\ -4& 3\end{array}\right]\left[\begin{array}{cc}1& -3\\ -4& 3\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}1+12& -3-6\\ -4-12& 12+9\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}13& -9\\ -16& 21\end{array}\right] \\ {A}^2-5A-14I \\ \Rightarrow A^2-3A-7I=\left[\begin{array}{cc}13& -9\\ -16& 21\end{array}\right]-3\left[\begin{array}{cc}1& -3\\ -4& 3\end{array}\right]-7\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow A^2-3A-7I=\left[\begin{array}{cc}13& -9\\ -16& 21\end{array}\right]-\left[\begin{array}{cc}3& -9\\ -12& 9\end{array}\right]-\left[\begin{array}{cc}7& 0\\ 0& 7\end{array}\right] \\ \Rightarrow A^2-3A-7I=\left[\begin{array}{cc}3& 0\\ -4& 5\end{array}\right] \end{gathered}$$