Prove that if $A$ is a square matrix with integer entries and $\det(A)=\pm 1$, then the inverse of $A$ contains all integer entries.
Prove the inverse of an integer matrix has only integer entries?
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linear-algebra
matrices
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0What do you know? What have you tried? – 2012-10-14
1 Answers
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Hint: Consider the inverse written in terms of the adjugate matrix.
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0@KushalBhuyan A general statement is that a matrix over $R$ is invertible (over $R$) if and only if $\det M$ is a unit of $R$. The only units of $R=\mathbb{Z}$ are $\pm1$, so this proves the statement you're looking for. More explicitly, we know that $\det M \cdot \det M^{-1} = 1$. Since $\det M$ and $\det M^{-1}$ are both integers (since $M$ and $M^{-1}$ are integer valued matrices), the only way for this to happen is if $\det M = \det M^{-1} = \pm 1$. – 2016-05-09