Matrices & Determinants
Symmetric Matrices
Grade 12
Question:
<p>Let <em>A</em> and <em>B</em> two symmetric matrices of order 3.<br><strong>Statement 1:</strong> <em>A</em>(<em>BA</em>) and (<em>AB</em>)<em>A</em> are symmetric matrices.<br><strong>Statement 2:</strong> <em>AB</em> is symmetric matrix if matrix multiplication of <em>A</em> with <em>B</em> is commutative.</p>
<p>Statement 1 is false, statement 2 is true.</p>
<p>Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.</p>
<p>Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.</p>
<p>Statement 1 is true, statement 2 is false.</p>
Step-by-Step Solution
Key Concept: A product of matrices is symmetric iff it equals its transpose; use (XY)ᵀ = YᵀXᵀ and the fact that Aᵀ = A and Bᵀ = B for symmetric matrices to verify which products are symmetric.
<p><strong>Step 1: Verify Statement 1 - Is A(BA) symmetric?</strong></p><p>[A(BA)]ᵀ = (BA)ᵀAᵀ = AᵀBᵀAᵀ = ABA (since Aᵀ=A, Bᵀ=B)</p><p>Need to check if ABA = A(BA). Since matrix multiplication is associative: A(BA) = (AB)A = ABA ✓</p><p>Therefore A(BA) is symmetric. Similarly (AB)A is also symmetric by same logic.</p><p><strong>Step 2: Verify Statement 2 - Is AB symmetric?</strong></p><p>(AB)ᵀ = BᵀAᵀ = BA (since Aᵀ=A, Bᵀ=B)</p><p>For AB to be symmetric: AB = (AB)ᵀ = BA</p><p>This means AB is symmetric if and only if AB = BA (A and B commute).</p><p><strong>Step 3: Assess both statements</strong></p><p>Statement 1: TRUE - Both A(BA) and (AB)A are symmetric matrices</p><p>Statement 2: TRUE - AB is symmetric iff AB and BA are equal (commutative)</p><p>Statement 2 correctly explains the condition for Statement 1.</p><p>∴ Answer: C (Both statements are true; Statement 2 is the correct explanation of Statement 1)</p>
Correct Answer: C