Question

If the matrix $A=\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]and{A}^2=kA$, then find the value of k.

Answer 1

$A=\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]$

$A^2=\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]$

$A^2=\left[\begin{array}{cc}4+4& -4-4\\ -4-4& 4+4\end{array}\right]=\left[\begin{array}{cc}8& -8\\ -8& 8\end{array}\right]$

Now, $A^2=kA$

$\left[\begin{array}{cc}8& -8\\ -8& 8\end{array}\right]=k\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]$

Hence, $k=4$