Question

If \( P \) and \( Q \) are two \( 3 \times 3 \) matrices such that \( |PQ| = 1 \) and \( |P| = 9 \), then the determinant of adjoint of the matrix \( P . Adj \ 3Q \) is:

• A. \(9^4\)
• B. \(\frac{1}{9^4} \)
• C. \(9^2\)
• D. \(\frac{1}{9^2} \)

Hint:

For determinant-based problems, remember key properties such as \( \text{det}(\text{Adj } A) = (\text{det } A)^{n-1} \).

Answer 1

The Correct Option is A

Step 1: Using the Determinant Property For any \( n \times n \) matrix \( A \), the determinant of its adjugate matrix is given by: \[ \text{det}(\text{Adj } A) = (\text{det } A)^{n-1} \] Since \( P \) and \( Q \) are \( 3 \times 3 \) matrices, applying this formula: \[ \text{det}(\text{Adj } P) = (|P|)^{3-1} = (9)^2 = 81. \]
Step 2: Applying the Property to Product Matrices Since \( |PQ| = 1 \), and using determinant properties: \[ |P . \text{Adj } 3Q| = |P|^2 \cdot |3Q|^2 = 9^2 \cdot 9^2 = 9^4. \] Thus, the correct answer is: \[ \boxed{9^4} \]