<p>Given \(A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}\) and \(A\,\text{adj}\,A = AA^T\), find \(5a + b\).</p>
Step-by-Step Solution
Key Concept: Use the fundamental property adj(A)·A = |A|·I to relate the given condition A·adj(A) = AA^T to the determinant and transpose, then compare matrix entries to extract values of a and b.
<p><strong>Step 1:</strong> Recall that A·adj(A) = |A|·I, so from A·adj(A) = AA^T we get: |A|·I = AA^T</p><p><strong>Step 2:</strong> Compute |A| = (5a)(2) - (-b)(3) = 10a + 3b</p><p><strong>Step 3:</strong> Calculate AA^T where A^T = [5a, 3; -b, 2]:</p><p>AA^T = [25a² + b², 15a - 2b; 15a - 2b, 9 + 4]</p><p>So AA^T = [25a² + b², 15a - 2b; 15a - 2b, 13]</p><p><strong>Step 4:</strong> From |A|·I = AA^T, we need:</p><p>(10a + 3b)·I = [25a² + b², 15a - 2b; 15a - 2b, 13]</p><p><strong>Step 5:</strong> This gives us:</p><p>• Diagonal: 10a + 3b = 25a² + b² AND 10a + 3b = 13</p><p>• Off-diagonal: 15a - 2b = 0, so b = 7.5a</p><p><strong>Step 6:</strong> From 10a + 3b = 13 and b = 7.5a:</p><p>10a + 3(7.5a) = 13 → 10a + 22.5a = 13 → 32.5a = 13 → a = 0.4</p><p>Therefore b = 7.5(0.4) = 3</p><p><strong>Step 7:</strong> Verify: 25(0.16) + 9 = 4 + 9 = 13 ✓ and 10(0.4) + 3(3) = 4 + 9 = 13 ✓</p><p>∴ 5a + b = 5(0.4) + 3 = 2 + 3 = <strong>5</strong></p>
Correct Answer: C