Question

If matrix $A=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$ and $A^2=KA$, then write the value of $K.$

Answer 1

If $A=\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$ then $A^2$ can be computed as shown below:

$A^2=\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)=\left(\begin{array}{cc}1\times 1+-1\times -1& 1\times -1+(-1)\times 1\\ -1\times 1+1\times -1& -1\times -1+1\times 1\end{array}\right)$

implies $A^2=\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$

Since $A^2=KA$, therefore,

$\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)=K\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$

$\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)=\left(\begin{array}{cc}K& -K\\ -K& K\end{array}\right)$

Now equating the matrices we get $K=2$.

Hence, $K=2$.