Matrices & Determinants
Matrix Operations
Grade 12
Question:
<p>If <strong>AB</strong> = <strong>A</strong> and <strong>BA</strong> = <strong>B</strong>, then which of the following is/are true?</p>
<p>(a) <strong>A</strong><sup>2</sup><strong>B</strong> = <strong>A</strong><sup>2</sup></p>
<p>(b) <strong>B</strong><sup>2</sup><strong>A</strong> = <strong>B</strong><sup>2</sup></p>
<p>(c) <strong>ABA</strong> = <strong>A</strong></p>
<p>(d) <strong>BAB</strong> = <strong>B</strong></p>
Step-by-Step Solution
Key Concept: Use the given conditions AB = A and BA = B to substitute and simplify products of matrices.
<p><strong>Step 1:</strong> Given: <strong>AB</strong> = <strong>A</strong> and <strong>BA</strong> = <strong>B</strong></p><p><strong>Step 2:</strong> Check option (a): <strong>A</strong><sup>2</sup><strong>B</strong> = <strong>A</strong>(<strong>AB</strong>) = <strong>A</strong> · <strong>A</strong> = <strong>A</strong><sup>2</sup> ✓</p><p><strong>Step 3:</strong> Check option (b): <strong>B</strong><sup>2</sup><strong>A</strong> = <strong>B</strong>(<strong>BA</strong>) = <strong>B</strong><strong>B</strong> = <strong>B</strong><sup>2</sup> ✓</p><p><strong>Step 4:</strong> Check option (c): <strong>ABA</strong> = <strong>A</strong>(<strong>BA</strong>) = <strong>AB</strong> = <strong>A</strong> ✓</p><p><strong>Step 5:</strong> Check option (d): <strong>BAB</strong> = <strong>B</strong>(<strong>AB</strong>) = <strong>BA</strong> = <strong>B</strong> ✓</p><p>∴ All options (a), (b), (c), (d) are correct.</p>
Correct Answer: a,b,c,d