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?

Right. A square matrix over $;\Bbb Z/6\Bbb Z;$ is invertible iff its determinant equals $;\pm1\pmod6;$ — DonAntonio, Feb 10 '18 at 18:13

Asinomás · Answer 1 · 2018-02-10T18:20:42.237

3

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 $\mathbb Z_6$

edited Feb 10 '18 at 18:20

answered Feb 10 '18 at 18:15

Asinomás

1 Answers1