if for matrix $A, m$ is eigen value and x is corresponding eigen vector then $Ax = mx$
$Ax$ $=$ $mx$ $\Rightarrow$ $\left (A^{2} \right )$$\times$ $=$ $\left (m^{2} \right )$$\times$
$Ax$ = $mx$ $\Rightarrow$ $\left ( nA \right )$$\times$ $=$ $\left ( nm \right )$$\times$for any real $n$
Now, finding ($a^{2}$ - $3$$A$ + $4$$I$$\times 1$ and $a^{2}$ - $3$$A$ + $4$$I$$\times 2$ will not be difficult.
Answer will be $(A)$.