Which of the following holds true for matrix multiplication?
follows commutative property
follows distributive property
follows associative property
None of these
Step-by-Step Solution
Key Concept: Matrix multiplication does not commute (AB ≠ BA generally), but it satisfies both the distributive property over addition and the associative property. Understanding which algebraic properties hold for matrices is fundamental to matrix algebra.
<p><strong>Step 1: Check Commutative Property (Option A)</strong></p><p>For matrices A and B, we need to verify if AB = BA.</p><p>Counterexample: Let A = [1 0; 0 0] and B = [0 1; 0 0]</p><p>AB = [0 1; 0 0], but BA = [0 0; 0 0]</p><p>Since AB ≠ BA, matrix multiplication is <strong>NOT commutative</strong>. ✗</p><p><strong>Step 2: Check Distributive Property (Option B)</strong></p><p>We verify if A(B + C) = AB + AC and (A + B)C = AC + BC.</p><p>Let A, B, C be matrices of compatible dimensions.</p><p>A(B + C) = A·(B + C) applies matrix multiplication over the sum, which equals AB + AC by the distributive axiom of matrix operations.</p><p>Similarly, (A + B)C = AC + BC.</p><p>These identities hold by definition of matrix addition and multiplication rules. ✓</p><p><strong>Step 3: Check Associative Property (Option C)</strong></p><p>We verify if (AB)C = A(BC).</p><p>For any three matrices A (m×n), B (n×p), C (p×q):</p><p>(AB)C: First compute AB (m×p), then multiply by C → result is m×q</p><p>A(BC): First compute BC (n×q), then multiply by A → result is m×q</p><p>By the associative property of scalar multiplication and summation in the component-wise computation of matrix products, (AB)C = A(BC) always holds. ✓</p><p><strong>∴ Answer: B, C</strong></p>
Correct Answer: B, C