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Let A be a square matrix whose entries are all integers. Which of the following statements is correct?

1. If det A = ±1, then A⁻¹ exists, but its entries are not necessarily all integers
2. If det A ≠ ±1, then A⁻¹ exists and every entry of A⁻¹ is non-integer
3. If det A = ±1, then A⁻¹ exists and all of its entries are integers
4. If det A = ±1, then A⁻¹ may fail to exist

Correct answer: If det A = ±1, then A⁻¹ exists and all of its entries are integers

Solution

For an integer matrix A, A^(-1) = adj(A)/det(A). The adjugate (cofactor) entries are integers, so when det A = +/-1 the inverse exists and every entry is an integer.

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