<p>If <span>A = </span><span>\begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix}</span>, then <span>\text{adj}(\text{adj } A)</span> is equal to</p>
Step-by-Step Solution
Key Concept: The formula for adjoint of adjoint is \(\text{adj}(\text{adj } A) = |A|^{n-2} A\). When determinant equals 1, this simplifies to A itself.
<p><strong>Step 1:</strong> Use the property \(\text{adj}(\text{adj } A) = |A|^{n-2} A\) when \(|A| \neq 0\)</p><p><strong>Step 2:</strong> Calculate the determinant of A:</p><p>\[|A| = \begin{vmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{vmatrix}\]</p><p>\[= 3(-3 + 4) + 3(2 - 0) + 4(-2 - 0) = 3(1) + 3(2) + 4(-2) = 3 + 6 - 8 = 1\]</p><p><strong>Step 3:</strong> Since \(|A| = 1 \neq 0\), matrix A is non-singular.</p><p><strong>Step 4:</strong> Apply the formula with \(n = 3\) (order of matrix):</p><p>\[\text{adj}(\text{adj } A) = |A|^{3-2} \cdot A = A^1 \cdot A = A\]</p><p>∴ Answer is <strong>(a) A</strong></p>
Correct Answer: A