Matrices & Determinants
Operations on Matrices
Grade 12
Question:
<p>If \(A\) and \(B\) are square matrices of size \(n \times n\) such that \(A^2 - B^2 = (A - B)(A + B)\), then which of the following will be always true?</p>
<p>\(A = B\)</p>
<p>\(AB = BA\)</p>
<p>Either \(A\) or \(B\) is a zero matrix</p>
<p>Either \(A\) or \(B\) is an identity matrix</p>
Step-by-Step Solution
Key Concept: Matrix multiplication is non-commutative, so A² - B² = (A - B)(A + B) only holds when AB = BA. The question asks what is ALWAYS true, meaning we must identify the necessary condition implied by the given equation.
<p><strong>Step 1:</strong> Expand (A - B)(A + B) for matrices:</p><p>(A - B)(A + B) = A² + AB - BA - B²</p><p><strong>Step 2:</strong> Compare with A² - B²:</p><p>For A² - B² = (A - B)(A + B), we need:</p><p>A² - B² = A² + AB - BA - B²</p><p><strong>Step 3:</strong> Simplify by canceling A² and B² from both sides:</p><p>0 = AB - BA</p><p>Therefore: AB = BA</p><p><strong>Step 4:</strong> This means A and B must commute (be commutative matrices) for the given equation to hold.</p><p>∴ Answer: B (which should state that AB = BA or A and B commute)</p>
Correct Answer: B