<p>If \(A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}\), then \(\text{adj}(3A^2 + 12A)\) is equal to</p>
<p>\(\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 72 & -84 \\ -63 & 51 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 51 & 84 \\ 63 & 72 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: Factor the matrix expression as 3A(A + 4I), then use the property adj(kB) = k^(n-1)·adj(B) for n×n matrices, combined with adj(BC) = adj(C)·adj(B). Finally, recognize that adj(A + 4I) simplifies using the Cayley-Hamilton theorem or direct calculation.
<p><strong>Step 1:</strong> Calculate the characteristic polynomial of A. det(A - λI) = (2-λ)(1-λ) - 12 = λ² - 3λ - 10 = (λ-5)(λ+2)</p><p><strong>Step 2:</strong> By Cayley-Hamilton theorem: A² - 3A - 10I = 0, so A² = 3A + 10I</p><p><strong>Step 3:</strong> Substitute into the given expression: 3A² + 12A = 3(3A + 10I) + 12A = 9A + 30I + 12A = 21A + 30I = 3(7A + 10I)</p><p><strong>Step 4:</strong> Calculate 7A + 10I = 7·[2, -3; -4, 1] + 10I = [14, -21; -28, 7] + [10, 0; 0, 10] = [24, -21; -28, 17]</p><p><strong>Step 5:</strong> Find det(7A + 10I) = 24(17) - (-21)(-28) = 408 - 588 = -180</p><p><strong>Step 6:</strong> For 2×2 matrices: adj(7A + 10I) = [17, 21; 28, 24]</p><p><strong>Step 7:</strong> adj(3(7A + 10I)) = 3^(2-1)·adj(7A + 10I) = 3·[17, 21; 28, 24] = [51, 63; 84, 72]</p><p>∴ Answer: D</p>
Correct Answer: D