Let A be a square matrix all of whose entries are integers

• Matrices Determinants

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

1. If det A = ± 1, then A^-1 exists and all its entries are integers
2. If det A = ± 1, then A^-1 exists and all its entries are non-integers
3. If det A = ± 1, then A^-1 exists and all its entries are not necessarily integers
4. If det A = ± 1, then A^-1 need not exist

Answer

Each entry of A is an integer, so the co-factor of every entry is an integer.

Then, each entry of adjoint is integer.

Also det A = ± 1 and we know that

A^-1 = 1/detA (adj A)

The correct option is A.