Matrices & Determinants
Properties of Matrices
Grade 12
Question:
<p>If <em>A</em> and <em>B</em> are two square matrices such that \(B = -A^{-1}BA\), then \((A + B)^2\) is equal to</p>
<p>\(A^2 + B^2\)</p>
<p>\(O\)</p>
<p>\(A^2 + 2AB + B^2\)</p>
<p>\(A + B\)</p>
Step-by-Step Solution
Key Concept: The condition B = -A⁻¹BA can be rearranged to show that AB + BA = 0, which means A and B anti-commute. This special relationship allows (A+B)² to simplify dramatically.
<p><strong>Step 1:</strong> Start with the given condition: B = -A⁻¹BA</p><p><strong>Step 2:</strong> Multiply both sides by A on the left: AB = -BA</p><p>This gives us: <strong>AB + BA = 0</strong></p><p><strong>Step 3:</strong> Expand (A+B)² using the definition:</p><p>(A+B)² = (A+B)(A+B) = A² + AB + BA + B²</p><p><strong>Step 4:</strong> Substitute AB + BA = 0:</p><p>(A+B)² = A² + (AB + BA) + B² = A² + 0 + B²</p><p><strong>Step 5:</strong> Therefore: <strong>(A+B)² = A² + B²</strong></p><p>∴ Answer: A</p>
Correct Answer: A