Question

If $A=\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right]$, find k such that A² = kA − 2I₂

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right] \\ \mathrm{Now}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right]\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}9-8& -6+4\\ 12-8& -8+4\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}1& -2\\ 4& -4\end{array}\right] \\ {A}^2=kA-2I_2 \\ \Rightarrow \left[\begin{array}{cc}1& -2\\ 4& -4\end{array}\right]=k\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right]-2\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}1& -2\\ 4& -4\end{array}\right]=\left[\begin{array}{cc}3k& -2k\\ 4k& -2k\end{array}\right]-\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}1& -2\\ 4& -4\end{array}\right]=\left[\begin{array}{cc}3k-2& -2k-0\\ 4k-0& -2k-2\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \therefore 1=3k-2 \\ \Rightarrow 1+2=3k \\ \Rightarrow 3=3k \\ \Rightarrow k=1 \end{gathered}$$