Matrices & Determinants
Idempotent Matrices
Grade 12
Question:
<p>If <strong>A</strong>, <strong>B</strong> and <strong>A</strong> + <strong>B</strong> are idempotent matrices, then <strong>AB</strong> is equal to</p>
<p>(a) <strong>BA</strong></p>
<p>(b) −<strong>BA</strong></p>
<p>(c) <strong>I</strong></p>
<p>(d) <strong>O</strong></p>
Step-by-Step Solution
Key Concept: Use the property of idempotent matrices (A² = A) and expand (A + B)² to find the relationship between AB and BA.
<p><strong>Solution:</strong></p><p>Since <strong>A</strong>, <strong>B</strong> and <strong>A</strong> + <strong>B</strong> are idempotent matrices:</p><p>$\mathbf{A}^2 = \mathbf{A}; \quad \mathbf{B}^2 = \mathbf{B}; \quad (\mathbf{A} + \mathbf{B})^2 = \mathbf{A} + \mathbf{B}$</p><p>Consider $(\mathbf{A} + \mathbf{B})^2 = \mathbf{A} + \mathbf{B}$</p><p>$\mathbf{A}^2 + \mathbf{AB} + \mathbf{BA} + \mathbf{B}^2 = \mathbf{A} + \mathbf{B}$</p><p>$\mathbf{A} + \mathbf{AB} + \mathbf{BA} + \mathbf{B} = \mathbf{A} + \mathbf{B}$</p><p>$\mathbf{AB} + \mathbf{BA} = \mathbf{O}$</p><p>$\therefore \mathbf{AB} = -\mathbf{BA}$</p>
Correct Answer: b