If A and B are non- singular square matrix of same order `3xx3`, then which of the following options is correct?

Options

A. `|adj(AB)|=|A||B|`
B. `|(adjAB)^(-1)|=|adj(AB)|`
C. `|adj(AB)^(-1)|=|adj(AB)^(-1)|`
D. `|adj(AB)^(T)|=|AB|^(-2)`

Answer

To solve the problem, we need to analyze the properties of non-singular square matrices and their determinants, particularly focusing on the adjoint and inverse operations.

### Step-by-Step Solution:

1. **Understanding Non-Singular Matrices**:
- A non-singular matrix is one that has an inverse. For a matrix \( A \) of order \( n \), if \( \text{det}(A) \neq 0 \), then \( A \) is non-singular.

2. **Properties of Determinants**:
- For any two \( n \times n \) matrices \( A \) and \( B \):
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \]
- The determinant of the adjoint of a matrix \( A \) is given by:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \]
- For a \( 3 \times 3 \) matrix, this becomes:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^2 \]

3. **Analyzing the Options**:
- **Option A**: \( \text{adj}(AB) = \text{det}(A) \cdot \text{det}(B) \)
- This is incorrect because \( \text{adj}(AB) = \text{adj}(A) \cdot \text{adj}(B) \) and not equal to the product of the determinants.

- **Option B**: \( \text{det}(\text{adj}(AB^{-1})) = \text{adj}(A) \cdot \text{adj}(B) \)
- This is also incorrect. The correct relation involves the determinant of the adjoint and the determinant of the inverse, which leads to a different expression.

- **Option C**: \( \text{adj}(AB^{-1}) = \text{adj}(A) \cdot \text{adj}(B^{-1}) \)
- This is correct. The adjoint of the product of matrices is the product of their adjoints in reverse order.

4. **Conclusion**:
- After evaluating the options, we find that the correct statement is:
\[ \text{adj}(AB^{-1}) = \text{adj}(A) \cdot \text{adj}(B^{-1}) \]
- Therefore, the correct answer is **Option C**.