Matrix of Primes

\[ \large {\begin{bmatrix} 3 & 5 \\ 2 & 7 \end{bmatrix}} \]

Above is a \(2 \times 2\) matrix having elements from the set of first \(4\) prime numbers i-e \(S= \left\{ 2,3,5,7 \right\} \) such that they may or may not repeat.

For how many ordered tuples \(T(w,x,y,z)\), where \(w,x,y,z\) belongs to the set \(S\), determinant of the matrix \(M=\begin{bmatrix} w & y \\ x & z \end{bmatrix}\) is also prime ?


This problem is a part of set Prime Crimes via Computer Science
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