<p>If <i>A</i> is nonsingular and \((A - 2I)(A - 4I) = O\), then \(\frac{1}{6}A + \frac{4}{3}A^{-1}\) is equal to</p>
Step-by-Step Solution
Key Concept: Since (A - 2I)(A - 4I) = O and A is nonsingular, we can expand and rearrange to find A² in terms of A, then use this relation to express the given expression in terms of I.
<p><strong>Step 1:</strong> Expand (A - 2I)(A - 4I) = O</p><p>A² - 4A - 2A + 8I = O</p><p>A² - 6A + 8I = O</p><p><strong>Step 2:</strong> Rearrange: A² = 6A - 8I</p><p><strong>Step 3:</strong> Divide both sides by A (multiply by A⁻¹ on the right):</p><p>A = 6I - 8A⁻¹</p><p>8A⁻¹ = 6I - A</p><p>A⁻¹ = (3/4)I - (1/8)A</p><p><strong>Step 4:</strong> Compute (1/6)A + (4/3)A⁻¹</p><p>= (1/6)A + (4/3)[(3/4)I - (1/8)A]</p><p>= (1/6)A + I - (1/6)A</p><p>= I</p><p>∴ Answer: B (which equals I)</p>
Correct Answer: B