If `AneB, AB = BA and A^(2)=B^(2)`, then the value of the determinant of matrix `A+B` is (where A and B are square matrices of order `3xx3` )

Options

A. 0
B. 1
C. `3^(3)`
D. `3^(2)`

Answer

To solve the problem, we need to find the value of the determinant of the matrix \( A + B \) given the conditions \( A \neq B \), \( AB = BA \), and \( A^2 = B^2 \).

### Step-by-Step Solution:

1. **Start with the equation \( A^2 = B^2 \)**:
\[ A^2 - B^2 = 0 \]
This can be factored using the difference of squares:
\[ (A + B)(A - B) = 0 \]

2. **Analyze the product \( (A + B)(A - B) = 0 \)**:
Since the product of two matrices is zero, either \( A + B = 0 \) or \( A - B = 0 \) (or both). However, we know from the problem statement that \( A \neq B \), which means \( A - B \neq 0 \).

3. **Conclude \( A + B \) must be a null matrix**:
Since \( A - B \) is not the zero matrix, the only possibility left is:
\[ A + B = 0 \]
This implies:
\[ B = -A \]

4. **Find the determinant of \( A + B \)**:
Since \( A + B = 0 \), the determinant of the null matrix is:
\[ \det(A + B) = \det(0) = 0 \]

5. **Final Result**:
Therefore, the value of the determinant of the matrix \( A + B \) is:
\[ \boxed{0} \]