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 step-by-step, we will analyze the properties of the matrices involved and derive the required results.

### Given:
- \( B \) is an idempotent matrix, which means \( B^2 = B \).
- \( A = I - B \), where \( I \) is the identity matrix.

### Step 1: Find \( A^2 \)
We start by squaring \( A \):
\[ A^2 = (I - B)^2 \]
Using the formula for the square of a binomial, we expand this:
\[ A^2 = I^2 - 2IB + B^2 \]
Since \( I^2 = I \) and \( IB = B \), we can rewrite it as:
\[ A^2 = I - 2B + B^2 \]

### Step 2: Substitute \( B^2 \)
Since \( B \) is idempotent, we know that \( B^2 = B \). Substituting this into our equation gives:
\[ A^2 = I - 2B + B \]
This simplifies to:
\[ A^2 = I - B \]

### Step 3: Relate \( A^2 \) to \( A \)
Recall that \( A = I - B \), so we can substitute \( A \) back into the equation:
\[ A^2 = A \]
This shows that \( A \) is also an idempotent matrix.

### Step 4: Find \( AB \)
Next, we calculate \( AB \):
\[ AB = (I - B)B \]
Expanding this gives:
\[ AB = IB - B^2 \]
Again, using \( IB = B \) and \( B^2 = B \):
\[ AB = B - B = 0 \]
Thus, \( AB \) is the null matrix.

### Step 5: Find \( BA \)
Now, we calculate \( BA \):
\[ BA = B(I - B) \]
Expanding this gives:
\[ BA = BI - B^2 \]
Using \( BI = B \) and \( B^2 = B \):
\[ BA = B - B = 0 \]
Thus, \( BA \) is also the null matrix.

### Conclusion
From the steps above, we have derived the following results:
1. \( A^2 = A \) (A is idempotent)
2. \( AB = 0 \) (AB is the null matrix)
3. \( BA = 0 \) (BA is the null matrix)

### Final Answer:
Thus, the statements that are true are:
- \( A^2 = A \)
- \( AB = 0 \)
- \( BA = 0 \)