If adj`B=A ,|P|=|Q|=1,t h e na d j(Q^(-1)B P^(-1))` is `P Q` b. `Q A P` c. `P A Q` d. `P A^1Q`

Options

A. `PQ`
B. `QAP`
C. `PAQ`
D. `PA^(-1)Q`

Explanation

To solve the problem, we need to find the adjoint of the matrix expression \( Q^{-1} B P^{-1} \) given that \( \text{adj}(B) = A \) and \( |P| = |Q| = 1 \).

### Step-by-step Solution:

1. **Write the expression for the adjoint:**
\[ \text{adj}(Q^{-1} B P^{-1}) = \text{adj}(P^{-1}) \cdot \text{adj}(B) \cdot \text{adj}(Q^{-1}) \]

2. **Use the property of adjoint for inverses:**
The property states that:
\[ \text{adj}(A^{-1}) = \frac{\text{adj}(A)}{|A|} \]
Applying this to \( P^{-1} \) and \( Q^{-1} \):
\[ \text{adj}(P^{-1}) = \frac{\text{adj}(P)}{|P|} \quad \text{and} \quad \text{adj}(Q^{-1}) = \frac{\text{adj}(Q)}{|Q|} \]

3. **Substituting the determinants:**
Since \( |P| = 1 \) and \( |Q| = 1 \), we have:
\[ \text{adj}(P^{-1}) = \text{adj}(P) \quad \text{and} \quad \text{adj}(Q^{-1}) = \text{adj}(Q) \]

4. **Substituting the adjoint of B:**
We know from the problem statement that \( \text{adj}(B) = A \). Thus, we can substitute:
\[ \text{adj}(Q^{-1} B P^{-1}) = \text{adj}(P) \cdot A \cdot \text{adj}(Q) \]

5. **Final expression:**
We can write the final expression as:
\[ \text{adj}(Q^{-1} B P^{-1}) = P A Q \]

### Conclusion:
The answer is \( P A Q \).