Question

If matrix $A=\left[\begin{array}{cc}3& -3\\ -3& 3\end{array}\right]$ and $A^2=\lambda A$, then write the value of $\lambda$.

Answer 1

If $A=\left(\begin{array}{cc}3& -3\\ -3& 3\end{array}\right)$ then $A^2$ can be computed as shown below:

$A^2=\left(\begin{array}{cc}3& -3\\ -3& 3\end{array}\right)\left(\begin{array}{cc}3& -3\\ -3& 3\end{array}\right)=\left(\begin{array}{cc}3\times 3+-3\times -3& 3\times -3+(-3)\times 3\\ -3\times 3+3\times -3& -3\times -3+3\times 3\end{array}\right)$

implies $A^2=\left(\begin{array}{cc}18& -18\\ -18& 18\end{array}\right)$

Since $A^2=\lambda A$, therefore,

$\left(\begin{array}{cc}18& -18\\ -18& 18\end{array}\right)=\lambda \left(\begin{array}{cc}3& -3\\ -3& 3\end{array}\right)$

$\left(\begin{array}{cc}18& -18\\ -18& 18\end{array}\right)=\left(\begin{array}{cc}3\lambda & -3\lambda \\ -3\lambda & 3\lambda \end{array}\right)$

Now equating the matrices we get $3\lambda =18⟹\lambda =6$.