Matrices & Determinants
Invertible Matrices
Grade 12
Question:
<p>Suppose \(A\) is any \(3 \times 3\) non-singular matrix and \((A - 3I)(A - 5I) = O\), where \(I = I_3\) and \(O = O_3\). If \(\alpha A + \beta A^{-1} = 4I\), then \(\alpha + \beta\) is equal to</p>
<p>8</p>
<p>7</p>
<p>13</p>
<p>12</p>
Step-by-Step Solution
Key Concept: From (A - 3I)(A - 5I) = O, use the minimal polynomial to find that A satisfies a linear relation, then divide by A to create an equation involving both A and A⁻¹.
<p><strong>Step 1:</strong> Expand (A - 3I)(A - 5I) = O</p><p>A² - 5A - 3A + 15I = O</p><p>A² - 8A + 15I = O</p><p><strong>Step 2:</strong> Since A is non-singular, divide both sides by A (multiply on the right by A⁻¹)</p><p>A - 8I + 15A⁻¹ = O</p><p>A + 15A⁻¹ = 8I</p><p><strong>Step 3:</strong> Compare with αA + βA⁻¹ = 4I</p><p>Divide the equation from Step 2 by 2:</p><p>(1/2)A + (15/2)A⁻¹ = 4I</p><p><strong>Step 4:</strong> Match coefficients</p><p>α = 1/2 and β = 15/2</p><p>∴ α + β = 1/2 + 15/2 = 16/2 = <strong>8</strong></p>
Correct Answer: D