If A is a square matrix such that `A^(2)=A,` then `(I-A)^(2)+A=`

Options

A. `I`
B. `A`
C. `3A`
D. `I-A`

Answer

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

### Step-by-Step Solution:

1. **Expand \((I - A)^2\)**:
\[ (I - A)^2 = (I - A)(I - A) \]
Using the distributive property (matrix multiplication):
\[ = I \cdot I - I \cdot A - A \cdot I + A \cdot A \]
Since \(I \cdot I = I\) and \(A \cdot A = A\) (given \(A^2 = A\)):
\[ = I - A - A + A \]
Combining like terms:
\[ = I - 2A + A = I - A \]

2. **Add \(A\) to the result**:
Now we add \(A\) to the expanded expression:
\[ (I - A) + A = I - A + A \]
The \( -A\) and \( +A\) cancel each other out:
\[ = I \]

Thus, we find that:
\[ (I - A)^2 + A = I \]

### Final Answer:
The result is \(I\).