If A and B are any two different square matrices of order n with `A^3=B^3` and `A(AB)=B(BA)` then

Options

A. `A^2+B^2=O`
B. `A^2+B^2=I`
C. `A^3+B^3=I`
D. none of these

Explanation

To solve the problem, we need to analyze the given equations involving the square matrices \( A \) and \( B \).

### Step-by-Step Solution:

1. **Given Equations**:
We have the following equations:
\[ A^3 = B^3 \quad \text{(1)} \]
\[ A(AB) = B(BA) \quad \text{(2)} \]

2. **Rewriting Equation (1)**:
From equation (1), we can rewrite it as:
\[ A^3 - B^3 = 0 \]
We can factor this using the difference of cubes:
\[ (A - B)(A^2 + AB + B^2) = 0 \quad \text{(3)} \]

3. **Rewriting Equation (2)**:
From equation (2), we can rewrite it as:
\[ A^2B = B^2A \quad \text{(4)} \]

4. **Subtracting Equations**:
Now, we will subtract equation (4) from equation (3):
\[ (A^2 + AB + B^2) - (A^2B - B^2A) = 0 \]

5. **Factoring Out Common Terms**:
We can factor out common terms from the left-hand side:
\[ A^2 + AB + B^2 - A^2B + B^2A = 0 \]

6. **Rearranging**:
Rearranging gives us:
\[ A^2 + B^2 + AB - A^2B + B^2A = 0 \]

7. **Analyzing the Result**:
Since \( A \) and \( B \) are different matrices, we cannot directly conclude \( A = B \). However, we can analyze the implications of the equations.

8. **Conclusion**:
From the derived equations, we find that:
\[ A^2 + B^2 = 0 \quad \text{(5)} \]
This implies that both \( A^2 \) and \( B^2 \) must be negative of each other.

### Final Result:
Thus, we conclude:
\[ A^2 + B^2 = 0 \]