# Math Help - Matrix determinants (modulo)

1. ## Matrix determinants (modulo)

When calculating the inverse of a normal matrix, for $M = \begin{pmatrix}
x & y \\
z & v
\end{pmatrix}$
for example, you would simply use $\frac{1}{xv-yz}\begin{pmatrix}
v & -y \\
-z & x
\end{pmatrix}$
.

However, if your matrix was $\begin{pmatrix}
\bar{1} & \bar{2} \\
\bar{2} & \bar{0}
\end{pmatrix}$
in $\mathbb{Z}_3$, the inverse would be:

$(-\bar{4})^{-1}\begin{pmatrix}
\bar{0} & \bar{-2} \\
\bar{-2} & \bar{1}
\end{pmatrix} = (\bar{2})^{-1}\begin{pmatrix}
\bar{0} & \bar{1} \\
\bar{1} & \bar{1}
\end{pmatrix}$
.

Ok up to here, just forgotten how to calculate $(\bar{2})^{-1}$

2. I suppose with $\mathbb{Z}_3$ you mean the invertible elements of $\mathbb{Z}/3\mathbb{Z}$? (I've seen different notation for that).
Anyway, $(\overline{2})^{-1}=2$ since $2\cdot 2 = 4 \equiv 1$ mod 3