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: \[ \boxed{9^4} \]

Answer 2

To find the determinant of the adjoint of the matrix \(P \cdot \text{Adj } 3Q\), we need to follow these steps:

1. Given: \(|PQ| = 1\) and \(|P| = 9\). We know \(|A \cdot B| = |A| \cdot |B|\) for any square matrices \(A\) and \(B\). Thus, \(|PQ| = |P| \cdot |Q| = 1\).
2. Substituting the value, \(|P| \cdot |Q| = 1\), we get \(9 \cdot |Q| = 1\), which implies \(|Q| = \frac{1}{9}\).
3. Now, consider the expression \(P \cdot \text{Adj } 3Q\). We need to find the determinant of its adjoint: \(|\text{Adj}(P \cdot \text{Adj } 3Q)|\).
4. The property of determinants states \(|\text{Adj}(A)| = |A|^{n-1}\) where \(n\) is the order of the square matrix. Thus, for a 3x3 matrix \(A\), \(|\text{Adj}(A)| = |A|^2\).
5. Since \(P\) and \(3Q\) are both 3x3 matrices, we know:
   • \(|3Q| = 3^3 \cdot |Q| = 27 \cdot \frac{1}{9} = 3\).
   • \(|\text{Adj } 3Q| = |3Q|^2 = 3^2 = 9\).
6. Now calculate \(|P \cdot \text{Adj } 3Q|\):
   • \(|P \cdot \text{Adj } 3Q| = |P| \cdot |\text{Adj } 3Q| = 9 \cdot 9 = 81\).
7. Finally, we calculate \(|\text{Adj}(P \cdot \text{Adj } 3Q)|\):
   • Using \(|\text{Adj}(A)| = |A|^2\), we get \(|P \cdot \text{Adj } 3Q|^2 = 81^2 = 6561 = 9^4\).

Therefore, the determinant of the adjoint of \(P \cdot \text{Adj } 3Q\) is \(\mathbf{9^4}\).