If A and B are square matrices of order 2 such that `det(AB) = det(BA)`, then which one of the following is correct?

Options

A. A must be a unit matix.
B. B must be unit matrix.
C. Both A and B must be unit matrices.
D. A and B need not be unit matrices.

Answer

To solve the problem, we need to analyze the condition given: \( \text{det}(AB) = \text{det}(BA) \) for square matrices \( A \) and \( B \) of order 2.

### Step-by-step Solution:

1. **Understanding the Determinant Property**:
We know that for any two square matrices \( A \) and \( B \), the determinant of their product can be expressed as:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \]
and
\[ \text{det}(BA) = \text{det}(B) \cdot \text{det}(A) \]
Thus, we can rewrite the condition \( \text{det}(AB) = \text{det}(BA) \) as:
\[ \text{det}(A) \cdot \text{det}(B) = \text{det}(B) \cdot \text{det}(A) \]
This equation is always true and does not provide any new information.

2. **Exploring the Implications**:
The condition \( \text{det}(AB) = \text{det}(BA) \) does not imply that \( A \) and \( B \) must be unit matrices. However, we need to consider specific cases where this equality holds.

3. **Special Case of Unit Matrices**:
If both \( A \) and \( B \) are unit matrices (identity matrices), then:
\[ \text{det}(A) = 1 \quad \text{and} \quad \text{det}(B) = 1 \]
Therefore:
\[ \text{det}(AB) = 1 \cdot 1 = 1 \]
and
\[ \text{det}(BA) = 1 \cdot 1 = 1 \]
Hence, \( \text{det}(AB) = \text{det}(BA) \) holds true.

4. **Conclusion**:
While \( \text{det}(AB) = \text{det}(BA) \) can hold for other matrices as well, the problem specifically asks for a condition that guarantees this equality. The most straightforward case is when both matrices are unit matrices.

### Final Answer:
The correct option is that both \( A \) and \( B \) must be unit matrices.