(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A is an invertible integer matrix. Prove that if det A = 1 or det A = -1, then the inverse of A is also an integer matrix.

Also prove the reverse, if A-inverse is an integer matrix then its determinant is 1 or -1.

2. Relevant equations

I'm not too sure how to start here. My first instinct is to work backwards from some properties of determinants that I know of.

det A-inv = det A

det [A(A-inv)] = det(A)det(A-inv) = det(A) det(A) = det I = 1, so det A = 1 or -1.

Same can be proven for A-inv, det (A-inv) det (A-inv) = det I = 1, so det A-inv = 1 or -1.

3. The attempt at a solution

I'm not so sure how to approach this from here. Sorry if this is a stupid question, I am studying on my own, not in college and not many other resources to turn to.

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# Homework Help: Prove that the inverse of an integer matrix is also an integer matrix

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