GATE XE 2024 | Question: 5

Question

Let $A$ be a $3 \times 3$ matrix whose eigenvalues are $2,3,4$ and let $I$ be the identity matrix of order 3 . If
\[
A^{-1}=\frac{1}{2 k}\left(A^{2}-9 A\right)+\frac{13}{k} I
\]
for some integer $k \neq 0$, then the value of $k$ is ___________.