Matrices & Determinants
Adjoint and Inverse of a Matrix
Grade 12
Question:
<p>It is given that each entry of matrix \(A\) is an integer. Which of the following is necessarily true?</p>
<p>\(\det(A) = \pm 1\)</p>
<p>All entries of \(A^{-1}\) are integers if \(\det(A) = \pm 1\)</p>
<p>\(A^{-1}\) always exists</p>
<p>All entries of \(A^{-1}\) are fractions</p>
Step-by-Step Solution
Key Concept: The determinant of a matrix with integer entries is always an integer because det(A) is computed via integer linear combinations (products and sums) of integer entries. This property holds regardless of whether A is invertible or what its rank is.
<p><strong>Step 1:</strong> Recall that the determinant of any n×n matrix A is computed as a sum of products of entries:</p><p>det(A) = Σ(±a₁ᵢ₁·a₂ᵢ₂·...·aₙᵢₙ)</p><p><strong>Step 2:</strong> If each entry aᵢⱼ is an integer, then every product of entries is an integer, and every sum/difference of integers is an integer.</p><p><strong>Step 3:</strong> Therefore, det(A) must be an integer.</p><p><strong>Step 4:</strong> This is true regardless of whether A is singular, invertible, or has any particular rank.</p><p>∴ <strong>Answer: B</strong> (The determinant of A is an integer)</p>
Correct Answer: B