Magic Matrix

Algebra Level 5

\[ \begin{bmatrix}{a} && {b} && {c} \\ {d} && {e} && {f} \\ {g} && {h} && {i}\end{bmatrix} \]

You are given a matrix \(S\) as shown above whose elements are denoted as 9 letters from \(a\) to \(i\), and each letter is a distinct positive integer between 1 to 9. If this matrix is not invertible and it satisfies the constraints \(a+d=b+e=c+f=g+h=i\) and \(a-b=d-c=1\), what is the value of 9-digit integer \(\overline{ abcdefghi}\)?

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