<p>If \(A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}\) and \(A \cdot \text{adj}\, A = AA^T\), then \(5a + b\) is equal to</p>
Step-by-Step Solution
Key Concept: Use the property that A·adj(A) = |A|·I, then equate it to AA^T to extract relationships between a and b through determinant and matrix multiplication.
<p><strong>Step 1:</strong> Find adj(A) for A = [5a, -b; 3, 2]</p><p>For 2×2 matrix, adj(A) = [2, b; -3, 5a]</p><p><strong>Step 2:</strong> Use property A·adj(A) = |A|·I</p><p>|A| = 5a(2) - (-b)(3) = 10a + 3b</p><p>So A·adj(A) = (10a + 3b)I = [10a+3b, 0; 0, 10a+3b]</p><p><strong>Step 3:</strong> Calculate AA^T where A^T = [5a, 3; -b, 2]</p><p>AA^T = [5a, -b; 3, 2]·[5a, 3; -b, 2]</p><p>= [25a² + b², 15a - 2b; 15a - 2b, 9 + 4]</p><p>= [25a² + b², 15a - 2b; 15a - 2b, 13]</p><p><strong>Step 4:</strong> Equate A·adj(A) = AA^T</p><p>Diagonal: 10a + 3b = 25a² + b² and 10a + 3b = 13</p><p>Off-diagonal: 0 = 15a - 2b</p><p><strong>Step 5:</strong> From 15a - 2b = 0, we get b = 7.5a</p><p>Substituting in 10a + 3b = 13:</p><p>10a + 3(7.5a) = 13</p><p>10a + 22.5a = 13</p><p>32.5a = 13 → a = 0.4, b = 3</p><p>∴ 5a + b = 5(0.4) + 3 = 2 + 3 = <strong>5</strong></p>
Correct Answer: A