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Let $A$ be a $n \times n$ matrix, $n$ even, with even diagonal elements and all other elements odd integers. Prove that $\det(A)\neq 0$. Can anyone give me a hint? Thank you.

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Use Laplace's formula to prove with induction, that $\det A$ is odd. – Tomas Aug 15 '13 at 8:10

Hint: Compute the determinant of $A$ reduced modulo $2$.

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The determinant is an odd integer then.

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