Matrices & Determinants
Symmetric and Skew-Symmetric Matrices
Grade 12
Question:
<p>Let <em>A</em> and <em>B</em> be symmetric matrices of same order. Then <em>AB – BA</em> is</p>
<p>A symmetric matrix</p>
<p>A skew-symmetric matrix</p>
<p>A null matrix</p>
<p>An identity matrix</p>
Step-by-Step Solution
Key Concept: The commutator [A,B] = AB - BA of two symmetric matrices is always skew-symmetric, meaning (AB - BA)ᵀ = -(AB - BA). This follows because (AB - BA)ᵀ = Bᵀ Aᵀ - Aᵀ Bᵀ = BA - AB = -(AB - BA).
<p><strong>Step 1:</strong> Given that A and B are symmetric matrices, so A = A<sup>ᵀ</sup> and B = B<sup>ᵀ</sup>.</p><p><strong>Step 2:</strong> Find the transpose of (AB - BA):<br/>(AB - BA)<sup>ᵀ</sup> = (AB)<sup>ᵀ</sup> - (BA)<sup>ᵀ</sup></p><p><strong>Step 3:</strong> Apply transpose property (XY)<sup>ᵀ</sup> = Y<sup>ᵀ</sup>X<sup>ᵀ</sup>:<br/>(AB)<sup>ᵀ</sup> = B<sup>ᵀ</sup>A<sup>ᵀ</sup> = BA (since A<sup>ᵀ</sup> = A and B<sup>ᵀ</sup> = B)<br/>(BA)<sup>ᵀ</sup> = A<sup>ᵀ</sup>B<sup>ᵀ</sup> = AB</p><p><strong>Step 4:</strong> Therefore:<br/>(AB - BA)<sup>ᵀ</sup> = BA - AB = -(AB - BA)</p><p><strong>Step 5:</strong> Since (AB - BA)<sup>ᵀ</sup> = -(AB - BA), the matrix AB - BA is skew-symmetric.</p><p>∴ Answer: B (AB - BA is skew-symmetric)</p>
Correct Answer: B