Let $M$ be a vector space of all $3\times 3$ real matrices and let $$A=\begin{pmatrix}2&3&1\\0&2&0\\0&0&3\end{pmatrix}.$$
I am struck on the following problem: Show that $\{X\in M:\text{det}(AX)=0\}$ is not a subspace of M.
Can you please let me know why it can't be a subspace of M? What would be the contradiction?