Question

If $A=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, satisfies the matrix equation A² = kA, write the value of k.

Answer 1

$$\begin{gathered} A^2=AA \\ \Rightarrow A^2=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}1+1& -1-1\\ -1-1& 1+1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right] \\ \mathrm{Now}, \\ {A}^2=kA \\ \Rightarrow \left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]=k\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]=\left[\begin{array}{cc}k& -k\\ -k& k\end{array}\right] \\ \therefore k=2 \end{gathered}$$