# 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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