If $A=\begin{bmatrix}1 & 0\\-1 & 5\end{bmatrix}$ and $I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ then the value of k so that $A^2=6A+kI$ is given by :

-5

The correct answer is Option (2) → -5

$A^2=A.A=\begin{bmatrix}1 & 0\\-1 & 5\end{bmatrix}\begin{bmatrix}1 & 0\\-1 & 5\end{bmatrix}=\begin{bmatrix}1 & 0\\-6 & 25\end{bmatrix}$

$A^2=6A+kI$

so $kI=A^2-6A=\begin{bmatrix}1 & 0\\-6 & 25\end{bmatrix}-\begin{bmatrix}6 & 0\\-6 & 30\end{bmatrix}$

$kI=\begin{bmatrix}-5 & 0\\0 & -5\end{bmatrix}⇒k=-5$