If B is an idempotent matrix, and `A=I-B`, then

Options

A. `A^(2)=A`
B. `A^(2)=I`
C. `AB=O`
D. `BA=O`

Explanation

To solve the problem, we need to analyze the properties of the matrices given that \( B \) is an idempotent matrix and \( A = I - B \).

### Step-by-Step Solution:

1. **Understanding Idempotent Matrix**:
An idempotent matrix \( B \) satisfies the property:
\[ B^2 = B \]

2. **Define Matrix A**:
Given that:
\[ A = I - B \]

3. **Calculate \( A^2 \)**:
To find \( A^2 \), we calculate:
\[ A^2 = (I - B)(I - B) \]
Expanding this using the distributive property:
\[ A^2 = I \cdot I - I \cdot B - B \cdot I + B \cdot B \]
This simplifies to:
\[ A^2 = I - B - B + B^2 \]
Since \( B^2 = B \) (because \( B \) is idempotent), we substitute \( B \) for \( B^2 \):
\[ A^2 = I - 2B + B \]
Thus:
\[ A^2 = I - B \]
Since \( A = I - B \), we have:
\[ A^2 = A \]

4. **Conclusion for Option 1**:
Therefore, we conclude that:
\[ A^2 = A \]
This means option 1 is correct.

5. **Calculate \( AB \)**:
Next, we calculate \( AB \):
\[ AB = (I - B)B \]
Expanding this gives:
\[ AB = IB - BB \]
Since \( IB = B \) and \( BB = B^2 = B \):
\[ AB = B - B = 0 \]

6. **Calculate \( BA \)**:
Now, we calculate \( BA \):
\[ BA = B(I - B) \]
Expanding this gives:
\[ BA = BI - B^2 \]
Again, since \( BI = B \) and \( B^2 = B \):
\[ BA = B - B = 0 \]

7. **Conclusion for Options 3 and 4**:
Thus, we find:
\[ AB = 0 \quad \text{and} \quad BA = 0 \]
This means options 3 and 4 are also correct.

### Final Answer:
- Option 1: \( A^2 = A \) (Correct)
- Option 2: \( A^2 = I \) (Incorrect)
- Option 3: \( AB = 0 \) (Correct)
- Option 4: \( BA = 0 \) (Correct)