Question

Can someone check to see if I did this proof correctly?
Let A be an nxn matrix with integer entries. Prove that A is nonsingular and A^-1 has integer entries if and only if det(A) plus/minus 1

Answer 1

Given: Matrix A is an nxn matrix with integer entries.
Proof: Since we are given a biconditional statement, we need to prove two claims:
1. If A is nonsingular and A^-1 has integer entries, then det(A) plus minus 1
2. If det(A) = plus/minus 1, then A is nonsingular and A^-1 has integer entries

Answer 2

For the first claim, A is nonsingular and A^-1 = adj(A)/det(A) has integer entries. If A has integer entries, then the cofactors of A are integers and adj(A) has only integer entries. Therefore 1/det(A) multiplied by each entry of adj(A) must be an integer. FUrthermore, if adj(A) has integer entries, then 1/det(A) must be an integer. As a result, det(A) = plus minus 1

Answer 3

FOr the second claim, det(A) plus minus 1 occurs when A is nonsingular and A^-1 = plus minus adj(A) have integer entries