Question

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

• A. If \( \text{det} (A) = \pm 1 \), then \( A^{-1} \) exists but all its entries are not necessarily integers.
• B. If \( \text{det} (A) \neq \pm 1 \), then \( A^{-1} \) exists and all its entries are non-integers.
• C. If \( \text{det} (A) = \pm 1 \), then \( A^{-1} \) exists and all its entries are integers.
• D. If \( \text{det} (A) = \pm 1 \), then \( A^{-1} \) need not exist.

Hint:

For matrices with integer entries, if \( \text{det}(A) = \pm 1 \), then the inverse \( A^{-1} \) will also have integer entries.

Answer 1

The Correct Option is C