Invertible matrix and inverse matrix

Question

Establish if the matrix $A\in M_3 (\Bbb{Z}_9)$

$$ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 0 \\ 0 & 3 & 6 \end{bmatrix} $$

is invertible, and if so, find the inverse matrix.

Answer 1

Since hint should be posted as answer, I post my hints here.

1. Add the second column to the third column. The bottom right corner vanishes.
2. To calculate $\det(A) \pmod 9$, expand along the last line. What's the cofactor matrix at the $(3,2)$-th entry?
3. Verifty that the determinant of the minor matrix $\det(M_{32}) \not\equiv 0 \pmod 3$, so $\det(A) \not\equiv 0 \pmod 9$