If A and B are two square matrices such that `AB=A` and `BA=B`, then `A^(2)` equals

Options

A. B
B. A
C. `I`
D. O

Explanation

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

### Step-by-step Solution:

1. **Start with the given equations**:
\[ AB = A \quad \text{(1)} \]
\[ BA = B \quad \text{(2)} \]

2. **From equation (1)**, we can manipulate it to express \( A \) in terms of \( B \):
\[ A = AB \quad \text{(from (1))} \]

3. **Now, substitute \( A \) from equation (1) into \( A^2 \)**:
\[ A^2 = A \cdot A = A \cdot (AB) \quad \text{(substituting A from (1))} \]

4. **Using the associative property of matrix multiplication**:
\[ A^2 = (A \cdot A)B = A(AB) \quad \text{(since A = AB)} \]

5. **Now, substitute \( AB \) back into the equation**:
\[ A^2 = A \cdot A = A \quad \text{(using (1))} \]

6. **Thus, we conclude**:
\[ A^2 = A \]

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