# 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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