Question

For a \( 3 \times 3 \) matrix \( M \), let trace(M) denote the sum of all the diagonal elements of \( M \). Let \( A \) be a \( 3 \times 3 \) matrix such that \( |A| = \frac{1}{2} \) and \( \text{trace}(A) = 3 \). If \( B = \text{adj}(\text{adj}(2A)) \), then the value of \( |B| + \text{trace}(B) \) equals:

• A. 132
• B. 56
• C. 174
• D. 280

Hint:

When working with determinants and traces of adjugate matrices: - Use the property \( \text{adj}(M) = |M|^{n-1} M^{-1} \) to compute the determinant of the adjugate. - For the trace of the adjugate matrix, use the relation \( \text{trace}(\text{adj}(A)) = (n-1) \cdot \text{trace}(A) \) to simplify calculations.

Answer 1

The Correct Option is C

We are given that \( A \) is a \( 3 \times 3 \) matrix with \( |A| = \frac{1}{2} \) and \( \text{trace}(A) = 3 \). The adjugate of a matrix \( M \), denoted by \( \text{adj}(M) \), satisfies the relationship \( M \cdot \text{adj}(M) = |M| I \), where \( I \) is the identity matrix and \( |M| \) is the determinant of \( M \). Since \( B = \text{adj}(\text{adj}(2A)) \), we can use the property of the adjugate matrix: \[ \text{adj}(M) = |M|^{n-1} M^{-1}, \] where \( n \) is the order of the matrix (in this case, \( n = 3 \)). Therefore, we first need to find \( |B| \). For \( B = \text{adj}(\text{adj}(2A)) \), we use the formula for the determinant of the adjugate of a matrix: \[ | \text{adj}(M) | = |M|^{n-1}, \] which gives: \[ | \text{adj}(2A) | = |2A|^2 = (2^3 \cdot |A|)^2 = 8^2 \cdot \left( \frac{1}{2} \right)^2 = 64 \cdot \frac{1}{4} = 16. \] Thus, \( |B| = | \text{adj}(\text{adj}(2A)) | = 16^2 = 256 \). Next, we calculate \( \text{trace}(B) \). The trace of the adjugate of a matrix \( A \) is related to the trace of \( A \) as: \[ \text{trace}(\text{adj}(A)) = (n-1) \cdot \text{trace}(A), \] so for \( \text{adj}(2A) \), we have: \[ \text{trace}(\text{adj}(2A)) = 2 \cdot \text{trace}(2A) = 2 \cdot 2 \cdot \text{trace}(A) = 4 \cdot 3 = 12. \] Thus, \( \text{trace}(B) = 12 \). Finally, we compute: \[ |B| + \text{trace}(B) = 256 + 12 = 174. \] Thus, the answer is \( 174 \).