Matrices & Determinants
Symmetric and Skew-symmetric matrices
Grade 12
Question:
<p>If <em>A</em> is symmetric as well as skew-symmetric matrix, then <em>A</em> is</p>
<p>(1) diagonal matrix</p>
<p>(2) null matrix</p>
<p>(3) triangular matrix</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: A matrix that is simultaneously symmetric (A = A^T) and skew-symmetric (A = -A^T) must satisfy both conditions simultaneously, which forces every element to equal its own negative.
<p><strong>Step 1:</strong> If A is symmetric, then A = A<sup>T</sup></p><p><strong>Step 2:</strong> If A is skew-symmetric, then A = -A<sup>T</sup></p><p><strong>Step 3:</strong> For A to be both simultaneously: A = A<sup>T</sup> and A = -A<sup>T</sup></p><p><strong>Step 4:</strong> Therefore: A<sup>T</sup> = -A<sup>T</sup>, which gives 2A<sup>T</sup> = 0, so A<sup>T</sup> = 0</p><p><strong>Step 5:</strong> This means A = 0 (the null/zero matrix)</p><p>∴ <strong>Answer: B</strong> (A is a null/zero matrix)</p>
Correct Answer: B