Singular Matrices Only

A = [ 1 2 3 4 5 6 7 8 9 ] ; B = [ a b c d e f g h i ] ; C = [ 1 a 2 b 3 c 4 d 5 e 6 f 7 g 8 h 9 i ] 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 × 3 3\times 3 matrices A A & B B consist of distinct digits from 1-9, denoted as a a to i i in B B , and the determinant of each matrix equals to 0. On the other hand, the matrix C C is constructed by adding each of B's digits next to A's at the same position ( C C = 10 A A + B 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 b + d d + f f + h h = 4 e 4e and b e b|e
  • 1 a \overline{1a} \mid 9 i \overline{9i} ; 3 c \overline{3c} \mid 7 g \overline{7g}

What is the value of a b c d e f g h i \overline{abcdefghi} ?


The answer is 416357298.

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