If A is a square matrix such that `A^2-A+I =0`, then the inverse of A is

Options

A. I-A
B. A-I
C. A
D. A+I

Explanation

To solve the problem, we start with the given equation involving the square matrix \( A \):

1. **Given Equation**:
\[ A^2 - A + I = 0 \]

2. **Rearranging the Equation**:
We can rearrange this equation to isolate \( I \):
\[ A^2 - A = -I \]

3. **Factoring Out \( A \)**:
We can factor out \( A \) from the left-hand side:
\[ A(A - I) = -I \]

4. **Multiplying Both Sides by \(-1\)**:
To make the right-hand side positive, we multiply the entire equation by \(-1\):
\[ -A(A - I) = I \]

5. **Rearranging Again**:
This can be rewritten as:
\[ A(I - A) = I \]

6. **Identifying the Inverse**:
From the equation \( A(I - A) = I \), we can see that \( I - A \) acts as the inverse of \( A \):
\[ A^{-1} = I - A \]

Thus, the inverse of the matrix \( A \) is:
\[ \boxed{I - A} \]