Matrices & Determinants
Properties of Determinants
Grade 12
Question:
<p>Let \(A\) be a square matrix all of whose entries are integers. Then which one of the following is true?</p>
<p>If \(\det(A) = \pm 1\), then \(A^{-1}\) exists but all its entries are not necessarily integers.</p>
<p>If \(\det(A) \neq \pm 1\), then \(A^{-1}\) exists and all its entries are non-integers.</p>
<p>If \(\det(A) = \pm 1\), then \(A^{-1}\) exists and all its entries are integers.</p>
<p>If \(\det(A) = \pm 1\), then \(A^{-1}\) need not exist.</p>
Step-by-Step Solution
Key Concept: The determinant of an integer matrix is always an integer (since det(A) is computed using only addition, subtraction, and multiplication of integer entries). However, the inverse matrix A⁻¹ has entries that are rational numbers (fractions), not necessarily integers, unless det(A) = ±1.
<p><strong>Step 1:</strong> If A is a square integer matrix, then det(A) is an integer (determinant is a polynomial expression in the entries of A).</p><p><strong>Step 2:</strong> The inverse is given by A⁻¹ = (1/det(A)) × adj(A). Since adj(A) has integer entries (cofactors are integers), A⁻¹ has entries of the form (integer)/det(A), which are rational but not necessarily integers.</p><p><strong>Step 3:</strong> For A⁻¹ to have all integer entries, we need det(A) to divide every entry of adj(A), which requires det(A) = ±1.</p><p><strong>Step 4:</strong> Therefore: det(A) is always an integer, but A⁻¹ is generally not an integer matrix unless det(A) = ±1.</p><p>∴ Answer: C (The statement that det(A) must be an integer is true, or the statement about A⁻¹ requiring det(A) = ±1 for integer entries)</p>
Correct Answer: C