Thread: Proof about square matrix

I found this example in a book of practice problems and if someone could explain this to me, I would be really grateful:

If all entries of a square matrix A are integers and |A|= + or - 1, show that all entries of A^-1 are integers.

Originally Posted by buckaroobill

I found this example in a book of practice problems and if someone could explain this to me, I would be really grateful:

If all entries of a square matrix A are integers and |A|= + or - 1, show that all entries of A^-1 are integers.

Adjugate matrix - Wikipedia, the free encyclopedia

Thanks! So can I just say for the proof that...

If |A|= +-1, then A^-1 will be comprised of all integers because A is a matrix of cofactor determinants (which are all integers). A determinant of an integer matrix is just an integer because it is the sum of integer products.

yes