Matrices & Determinants
Symmetric and Skew-Symmetric Matrices
Grade 12
Question:
<p>Let \(A\) and \(B\) be two nonsingular square matrices, \(A^T\) and \(B^T\) are the transpose matrices of \(A\) and \(B\), respectively, then which of the following are correct?</p>
<p>\(B^T A B\) is symmetric matrix if \(A\) is symmetric</p>
<p>\(B^T A B\) is symmetric matrix if \(B\) is symmetric</p>
<p>\(B^T A B\) is skew-symmetric matrix for every matrix \(A\)</p>
<p>\(B^T A B\) is skew-symmetric matrix if \(A\) is skew-symmetric</p>
Step-by-Step Solution
Key Concept: Properties of determinants and transposes are multiplicative and preserve under transposition: det(AB) = det(A)·det(B) and det(A^T) = det(A). Combined with the fact that nonsingular matrices have non-zero determinants, we can verify which statements about products and transposes hold universally.
<p><strong>Step 1: Recall fundamental properties</strong></p><p>For nonsingular matrices A, B:</p><ul><li>det(A) ≠ 0 and det(B) ≠ 0</li><li>det(A^T) = det(A)</li><li>det(AB) = det(A)·det(B)</li><li>(AB)^T = B^T·A^T (reverse order)</li></ul><p><strong>Step 2: Check statement A – det(A^T·B) = det(A)·det(B)</strong></p><p>det(A^T·B) = det(A^T)·det(B) = det(A)·det(B) ✓ <strong>CORRECT</strong></p><p><strong>Step 3: Check statement B – (A·B)^T = A^T·B^T</strong></p><p>(A·B)^T = B^T·A^T ≠ A^T·B^T (incorrect order) ✗ <strong>INCORRECT</strong></p><p><strong>Step 4: Check statement C – (A^T)^T = A^T</strong></p><p>(A^T)^T = A ≠ A^T (unless A is symmetric) ✗ <strong>INCORRECT</strong></p><p><strong>Step 5: Check statement D – det((AB)^T) = det(A)·det(B)</strong></p><p>det((AB)^T) = det(AB) = det(A)·det(B) ✓ <strong>CORRECT</strong></p><p>∴ Answer: A, D</p>
Correct Answer: A,D