Question

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

(A) If det(A) = ± 1, then A^-1 exists but all its entries are not necessarily integers.

(B) If det(A) ≠ ± 1, then A^-1 exists and all its entries are nonintegers.

(C) If det(A) = ± 1, then A^-1 exists and all its entries are integers.

(D) If det(A) = ± 1, then A^-1 need not exist.

Answer 1

Answer is (C) If det(A) = ± 1, then A^-1 exists and all its entries are integers.

It is given that each entry of A is integer. Therefore, the cofactor of every entry is an integer and so each entry in the adjoint of matrix A is an integer. So

det A = ±1 and A^-1 =(1/det(A))(adj)

This implies that all entries in A^-1 are integers.