Let A be an invertible n times nmatrix with integer entries

Let A be an invertible (n times n)-matrix with integer entries. If det(A) = plusminus1, show that A^-1 has integer entries.

Solution

It is given A be an invertible matrix with integer entries

In a nxn inverse matrix an element is given by Aij/det(A)

If detA = 1 then the entry in the inverse matrix = Aij ( an integer)

Cofactor Aji is the determinant of a submatrix of A, and the determinant of a matrix with integer entries is an integer