If `A ,B ,A+I ,A+B` are idempotent matrices, then `A B` is equal to `B A` b. `-B A` c. `I` d. `O`

Options

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

Explanation

To solve the problem, we need to analyze the properties of the given idempotent matrices \( A \) and \( B \). An idempotent matrix \( M \) satisfies the condition \( M^2 = M \).

### Step-by-Step Solution:

1. **Understand Idempotent Matrices**:
- Since \( A \) and \( B \) are idempotent, we have:
\[ A^2 = A \quad \text{(1)} \]
\[ B^2 = B \quad \text{(2)} \]

2. **Idempotent Property of \( A + I \)**:
- The matrix \( A + I \) is also idempotent:
\[ (A + I)^2 = A + I \quad \text{(3)} \]
- Expanding this gives:
\[ A^2 + 2A + I = A + I \]
- Substituting \( A^2 = A \) from equation (1):
\[ A + 2A + I = A + I \]
- Simplifying this, we find:
\[ 3A = 0 \quad \Rightarrow \quad A = 0 \]

3. **Idempotent Property of \( A + B \)**:
- The matrix \( A + B \) is also idempotent:
\[ (A + B)^2 = A + B \quad \text{(4)} \]
- Expanding this gives:
\[ A^2 + B^2 + AB + BA = A + B \]
- Substituting \( A^2 = A \) and \( B^2 = B \) from equations (1) and (2):
\[ A + B + AB + BA = A + B \]
- Canceling \( A + B \) from both sides:
\[ AB + BA = 0 \]

4. **Rearranging the Equation**:
- From \( AB + BA = 0 \), we can rearrange it to find:
\[ AB = -BA \]

5. **Conclusion**:
- Therefore, we conclude that:
\[ AB = -BA \]
- The correct answer is option **b. \( -BA \)**.