Matrices & Determinants
Types of Matrices
Grade 12
Question:
<p>Let \(A\) and \(B\) be two symmetric matrices of order 3.</p><p><strong>Statement-1:</strong> \(A(BA)\) and \((AB)A\) are symmetric matrices.</p><p><strong>Statement-2:</strong> \(AB\) is symmetric matrix if matrix multiplication of \(A\) and \(B\) is commutative.</p>
<p>Statement-1 is true, Statement-2 is false.</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 false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: For a product of matrices to be symmetric, we need (XY)ᵀ = XY. Use the property (XY)ᵀ = YᵀXᵀ and the fact that A and B are symmetric (Aᵀ = A, Bᵀ = B) to verify symmetry of products. Also recognize that AB is symmetric if and only if AB = BA.
<p><strong>Step 1: Verify Statement-1</strong></p><p>For A(BA): [A(BA)]ᵀ = (BA)ᵀAᵀ = AᵀBᵀAᵀ = ABA (since A, B are symmetric)</p><p>So A(BA) is symmetric. Similarly, [(AB)A]ᵀ = AᵀBᵀAᵀ = ABA = (AB)A</p><p>So (AB)A is symmetric. <strong>Statement-1 is TRUE.</strong></p><p><strong>Step 2: Verify Statement-2</strong></p><p>For AB to be symmetric: (AB)ᵀ = AB ⟹ BᵀAᵀ = AB ⟹ BA = AB</p><p>So AB is symmetric ⟺ AB = BA (i.e., A and B commute)</p><p>Statement-2 says AB is symmetric "if" AB is commutative, which is the correct necessary and sufficient condition.</p><p><strong>Statement-2 is TRUE and correctly explains Statement-1.</strong></p><p>∴ Answer: C (Both statements are true; Statement-2 is the correct explanation of Statement-1)</p>
Correct Answer: C