<p>\(A\) and \(B\) are square matrices of order \(n\) such that \(A^2 - B^2 = (A-B)(A+B)\). Which of the following must be true?</p>
Step-by-Step Solution
Key Concept: Matrix multiplication is non-commutative, so A² - B² = (A-B)(A+B) only holds when AB = BA (matrices commute). The question tests whether students incorrectly assume the algebraic identity A² - B² = (A-B)(A+B) always works for matrices without the commutativity condition.
<p><strong>Step 1:</strong> Expand the right side: (A-B)(A+B) = A² + AB - BA - B²</p><p><strong>Step 2:</strong> For the given equation A² - B² = (A-B)(A+B) to hold, we need: A² - B² = A² + AB - BA - B²</p><p><strong>Step 3:</strong> This simplifies to: 0 = AB - BA, which means AB = BA</p><p><strong>Step 4:</strong> Therefore, matrices A and B must commute for the given identity to be true.</p><p><strong>Conclusion:</strong> The condition A² - B² = (A-B)(A+B) is true if and only if AB = BA (A and B commute).</p><p>∴ Answer: B (Assuming B states that A and B must commute)</p>
Correct Answer: B