For a matrix A, if `A^(2)=A` and `B=I-A` then `AB+BA +I-(I-A)^(2)` is equal to (where, I is the identity matrix of the same order of matrix A)

Options

A. B
B. A
C. AB
D. I

Answer

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

### Step-by-step Solution:

1. **Identify the matrices and properties:**
- We know \( A^2 = A \) (which implies \( A \) is idempotent).
- We define \( B = I - A \).

2. **Expand the expression:**
- We need to compute \( AB + BA + I - (I - A)^2 \).
- Start by expanding \( (I - A)^2 \):
\[ (I - A)^2 = I^2 - 2IA + A^2 = I - 2A + A = I - A. \]
- Thus, we have:
\[ I - (I - A)^2 = I - (I - A) = A. \]

3. **Substituting back into the expression:**
- Now substitute \( (I - A)^2 \) back into the original expression:
\[ AB + BA + I - (I - A)^2 = AB + BA + A. \]

4. **Calculate \( AB \) and \( BA \):**
- Calculate \( AB \):
\[ AB = A(I - A) = A - A^2 = A - A = 0. \]
- Calculate \( BA \):
\[ BA = (I - A)A = A - A^2 = A - A = 0. \]

5. **Combine the results:**
- Now substitute \( AB \) and \( BA \) back into the expression:
\[ AB + BA + A = 0 + 0 + A = A. \]

### Conclusion:
The final result of the expression \( AB + BA + I - (I - A)^2 \) is \( A \).