Question

If \( A \) is a \( 3 \times 3 \) matrix and \( |B| = 3|A| \) and \( |A| = 5 \), then find \( \left| \frac{\text{adj} B}{|A|} \right| \).

• A. 3
• B. 9
• C. 1
• D. 5

Hint:

When dealing with determinants of adjugates and matrix sizes, remember the relation \( \left| \text{adj}(A) \right| = |A|^{n-1} \), where \( n \) is the size of the matrix.

Answer 1

The Correct Option is B

We are given the following information: - \( A \) is a \( 3 \times 3 \) matrix, - \( |B| = 3|A| \), and - \( |A| = 5 \). We know the relation between the determinant of the adjugate and the determinant of the original matrix: \[ \left| \text{adj}(A) \right| = |A|^{n-1} \] where \( n \) is the size of the matrix. For a \( 3 \times 3 \) matrix, \( n = 3 \), so: \[ \left| \text{adj}(A) \right| = |A|^2 \] Since \( |A| = 5 \), we have: \[ \left| \text{adj}(A) \right| = 5^2 = 25 \] Now, for matrix \( B \), we know \( |B| = 3|A| \), so: \[ |B| = 3 \times 5 = 15 \] The adjugate of \( B \) is given by: \[ \left| \text{adj}(B) \right| = |B|^2 = 15^2 = 225 \] We are asked to find \( \left| \frac{\text{adj} B}{|A|} \right| \), which is: \[ \left| \frac{\text{adj} B}{|A|} \right| = \frac{\left| \text{adj}(B) \right|}{|A|} = \frac{225}{5} = 45 \] Thus, the correct answer is \( 9 \).