Let A and B be two ` 3 xx 3` real matrices such that `(A^(2) - B^(2))` is invertible matrix . If `A ^(5) = B^(5) and A^(3) B^(2) = A^(2) B^(3)` , then the value of the determinant of the matrix `A^(3) + B^(3)` is equal to :

Options

A. 2
B. 4
C. 1
D. 0

Answer

To solve the problem, we need to analyze the given conditions and derive the value of the determinant of the matrix \( A^3 + B^3 \).

### Step-by-Step Solution:

1. **Given Conditions**:
- \( A^5 = B^5 \) (Equation 1)
- \( A^2 B^2 = B^2 A^2 \) (Equation 2)
- \( A^2 - B^2 \) is invertible (Equation 3)

2. **Using the Identity for Cubes**:
We can use the identity for the sum of cubes:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
This means we need to find the determinant of \( A^3 + B^3 \).

3. **Finding the Determinant**:
We can express the determinant as:
\[ \det(A^3 + B^3) = \det((A + B)(A^2 - AB + B^2)) = \det(A + B) \cdot \det(A^2 - AB + B^2) \]

4. **Using the Given Conditions**:
From Equation 1, since \( A^5 = B^5 \), we can infer that \( A \) and \( B \) might have some relation. Specifically, if we assume \( A = B \), then both sides of the equations hold true trivially.

5. **Analyzing the Invertibility Condition**:
Since \( A^2 - B^2 \) is invertible, it implies that \( A^2 \neq B^2 \). This means \( A \) and \( B \) cannot be equal. However, they must still satisfy the other conditions.

6. **Using the Condition \( A^2 B^2 = B^2 A^2 \)**:
The commutativity condition \( A^2 B^2 = B^2 A^2 \) suggests that \( A^2 \) and \( B^2 \) can be simultaneously diagonalized or have some common eigenvalues.

7. **Conclusion**:
Since \( A^2 - B^2 \) is invertible, we can conclude that the determinant of \( A^3 + B^3 \) must equal zero. This is because the product of the determinants leads to a contradiction if \( \det(A + B) \) or \( \det(A^2 - AB + B^2) \) were non-zero.

Thus, the final result is:
\[ \det(A^3 + B^3) = 0 \]