If $A=\begin{bmatrix}0 & -1\\0 & 2\end{bmatrix} $ and $I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ and $A^2=3A+kI,$ then the value of k is :

No real value of k exists

The correct answer is Option (3) → No real value of k exists

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

$⇒A^2=\begin{bmatrix}0 & -2\\0 & 4\end{bmatrix}$

so $A^2=3A+kI$

$⇒kI=\begin{bmatrix}0 & -2\\0 & 4\end{bmatrix}-\begin{bmatrix}0 & -3\\0 & 6\end{bmatrix}$

so $kI=\begin{bmatrix}0 & 1\\0 & 2\end{bmatrix}$

so for no value of k the equation holds

⇒ No real/imaginary value of k exists