Matrices & Determinants
Symmetric and Skew Symmetric Matrices
Grade 12
Question:
<p>Let \(A\) and \(B\) be any two \(3 \times 3\) matrices. If \(A\) is symmetric and \(B\) is skew symmetric, then the matrix \(AB - BA\) is:</p>
<p>skew symmetric.</p>
<p>symmetric.</p>
<p>neither symmetric nor skew symmetric.</p>
<p>\(I\) or \(-I\), where \(I\) is an identity matrix.</p>
Step-by-Step Solution
Key Concept: Recognize that AB - BA is the commutator bracket. For symmetric A and skew-symmetric B, the product AB - BA must satisfy specific symmetry properties based on the definitions A^T = A and B^T = -B.
<p><strong>Step 1:</strong> Let A be symmetric, so A^T = A. Let B be skew-symmetric, so B^T = -B.</p><p><strong>Step 2:</strong> Compute the transpose of (AB - BA):<br/>(AB - BA)^T = (AB)^T - (BA)^T = B^T A^T - A^T B^T</p><p><strong>Step 3:</strong> Substitute the properties:<br/>= (-B)(A) - (A)(-B) = -BA + AB = -(AB - BA)</p><p><strong>Step 4:</strong> Since (AB - BA)^T = -(AB - BA), by definition AB - BA is a skew-symmetric matrix.</p><p>∴ Answer: <strong>B (Skew-symmetric matrix)</strong></p>
Correct Answer: B