[Math] Finding the value of k in a matrix

Question

I can't figure out how to solve this question:
Find the value(s) of $k$ such that
$$A=\begin{bmatrix} 3 &0 &3 \\ -6 &3 &3k \\ -3 &6 &3k^2 \end{bmatrix}$$
has an inverse.
Any help would be greatly appreciated, I've been working on it for a long time!

Answer 1

A matrix only has an inverse iff its determinant is non-zero.

$A=3\begin{bmatrix} 1 &0 &1 \\ -2 &1 &k \\ -1 &2 &k^2 \end{bmatrix}$

and the determinant is then $27(k^2-2k-3)=27(k-3)(k+1)$, and so $k$ can take any value other than $3$ and $-1$.