Matrices & Determinants
Symmetric and skew-symmetric matrices
Grade 12
Question:
<p>Let \(A\) and \(B\) be two square matrices of the same size such that \(AB^T + BA^T = O\). If \(A\) is a skew-symmetric matrix then \(BA\) is</p>
<p>a symmetric matrix</p>
<p>a skew-symmetric matrix</p>
<p>an orthogonal matrix</p>
<p>an invertible matrix</p>
Step-by-Step Solution
Key Concept: If AB^T + BA^T = O and A is skew-symmetric (A^T = -A), substitute A^T = -A into the given equation to derive a relationship between B and A, revealing the nature of BA.
<p><strong>Step 1:</strong> Given: AB<sup>T</sup> + BA<sup>T</sup> = O and A is skew-symmetric, so A<sup>T</sup> = -A</p><p><strong>Step 2:</strong> Substitute A<sup>T</sup> = -A into AB<sup>T</sup> + BA<sup>T</sup> = O:<br/>AB<sup>T</sup> + B(-A) = O<br/>AB<sup>T</sup> - BA = O<br/>AB<sup>T</sup> = BA</p><p><strong>Step 3:</strong> Take transpose of both sides:<br/>(AB<sup>T</sup>)<sup>T</sup> = (BA)<sup>T</sup><br/>BA<sup>T</sup> = A<sup>T</sup>B<sup>T</sup><br/>B(-A) = (-A)B<sup>T</sup><br/>-BA = -AB<sup>T</sup><br/>BA = AB<sup>T</sup></p><p><strong>Step 4:</strong> From Step 2: AB<sup>T</sup> = BA, and from Step 3: BA = AB<sup>T</sup>, these are consistent. Taking transpose of BA = AB<sup>T</sup>:<br/>(BA)<sup>T</sup> = (AB<sup>T</sup>)<sup>T</sup> = BA<sup>T</sup> = B(-A) = -BA<br/>Therefore (BA)<sup>T</sup> = -BA</p><p><strong>Step 5:</strong> This means BA is skew-symmetric.</p><p>∴ Answer: A (BA is skew-symmetric)</p>
Correct Answer: A