Non-zero Product Game

Algebra Level 5

\[\begin{bmatrix} 0 & a \\ b & c \end{bmatrix} \begin{bmatrix} d & 0 \\ e & f \end{bmatrix} = \begin{bmatrix} g & h \\ 0 & i \end{bmatrix} \begin{bmatrix} -70 & 0 \\ 31 & 12 \end{bmatrix}\]

The elements \(a\) to \(i\) in the matrix equation above denote distinct digits from \(1\) to \(9\) such that every matrix has at least one prime element with one matrix having a pair of twin primes.

If \(S = \begin{bmatrix}{a} && {b} && {c} \\ {d} && {e} && {f} \\ {g} && {h} && {i}\end{bmatrix}\), compute \(|\det(S)|\).

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