# 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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