Matrix invertible or not?

Question

Given the matrix $A\in M_3(\mathbb{Z_6})$

$$ \begin{matrix} 5 & 3 & 3 \\ 1 & 4 & 1 \\ 3 & 5 & 1 \\ \end{matrix} $$

Is it invertible?

$det(A)=4$ but $4$ is not invertible in $\mathbb{Z_6}$ so the matrix is not invertible, right?

Answer 1

The matrix is not invertible, if it had an inverse say $B$ we would have $1=\det(AB)=\det(A)\det(B)=4\det(B)$ which is impossible in $\mathbb Z_6$