If a squared matrix of prime numbers, has inverse in $Z/pZ$, then what characteristics does it have?

Question

If S denotes the set of all the prime numbers $p$ with the property that the matrix $\begin{bmatrix}19 & 31 & 0\\29 & 31 & 0 \\79 & 23 & 59\end{bmatrix}$ has an inverse in the field $\mathbb{Z}/p\mathbb{Z}$.Then what characteristics does it have ?And what are the elements does S contain ?

Answer 1

The set is countably infinite except $${59,31,2,5}$$and thse are precisely the characteristic. You get this by seeing the determinant of this matrix is non zero for every $p$ except these