If A and B are two matrices such that AB=B and BA=A, then `A^(2)+B^(2)` is equal to

Options

A. 2AB
B. 2BA
C. A+B
D. AB

Answer

To solve the problem, we need to find \( A^2 + B^2 \) given that \( AB = B \) and \( BA = A \).

### Step-by-step Solution:

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

2. **Finding \( B^2 \)**:
We can express \( B^2 \) using equation (1):
\[ B^2 = B \cdot B = B \cdot (AB) \quad \text{(substituting \( B \) from equation (1))} \]
This simplifies to:
\[ B^2 = (BA)B \quad \text{(since \( AB = B \))} \]
Now, using equation (2) where \( BA = A \):
\[ B^2 = AB \quad \text{(substituting \( BA \) from equation (2))} \]
Since \( AB = B \) from equation (1), we have:
\[ B^2 = B \]

3. **Finding \( A^2 \)**:
Similarly, we can express \( A^2 \) using equation (2):
\[ A^2 = A \cdot A = A \cdot (BA) \quad \text{(substituting \( A \) from equation (2))} \]
This simplifies to:
\[ A^2 = (AB)A \quad \text{(since \( BA = A \))} \]
Now, using equation (1) where \( AB = B \):
\[ A^2 = BA \quad \text{(substituting \( AB \) from equation (1))} \]
Since \( BA = A \) from equation (2), we have:
\[ A^2 = A \]

4. **Combining Results**:
Now we can find \( A^2 + B^2 \):
\[ A^2 + B^2 = A + B \]

Thus, the final result is:
\[ A^2 + B^2 = A + B \]

### Final Answer:
\[ A^2 + B^2 = A + B \]