If $A=\begin{pmatrix}1 &0\\-1 & 7 \end {pmatrix}, I=\begin{pmatrix}1 &0\\0 & 1 \end {pmatrix},$ and $A^2=8A+KI $ the value of K is :

-7

The correct answer is Option (2) → -7

$A=\begin{pmatrix}1 &0\\-1 & 7 \end {pmatrix}$

$A^2=\begin{pmatrix}1 &0\\-1 & 7 \end {pmatrix}\begin{pmatrix}1 &0\\-1 & 7 \end {pmatrix}=\begin{pmatrix}1 &0\\-8 & 49 \end {pmatrix}$

$A^2-8A=KI$

$\begin{pmatrix}1 &0\\-8 & 49 \end {pmatrix}-\begin{pmatrix}8 &0\\-8 & 56 \end {pmatrix}=KI$

$\begin{pmatrix}-7 &0\\0 & -7 \end {pmatrix}=KI⇒K=-7$