Singular Matrices Only

\(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} ; B = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} ; C = \begin{bmatrix} \overline{1a} & \overline{2b} & \overline{3c} \\ \overline{4d} & \overline{5e} & \overline{6f} \\ \overline{7g} & \overline{8h} & \overline{9i} \end{bmatrix}\)

The \(3\times 3\) matrices \(A\) & \(B\) consist of distinct digits from 1-9, denoted as \(a\) to \(i\) in \(B\), and the determinant of each matrix equals to 0. On the other hand, the matrix \(C\) is constructed by adding each of B's digits next to A's at the same position (\(C\)= 10\(A\) + \(B\)), creating new 2-digit numbers with the following conditions:

  • The matrix C is also non-invertible.
  • There are 3 prime members in C.
  • \(b\) + \(d\) + \(f\) + \(h\) = \(4e\) and \(b|e\)
  • \(\overline{1a}\) \(\mid\) \(\overline{9i}\) ; \(\overline{3c}\) \(\mid\) \(\overline{7g}\)

What is the value of \(\overline{abcdefghi}\)?

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