If `A^(2)-A+I=O` then inverse of A is

Options

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

Answer

To solve the equation \( A^2 - A + I = O \), where \( O \) is the null matrix and \( I \) is the identity matrix, we want to find the inverse of the matrix \( A \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ A^2 - A + I = O \]

2. **Rearrange the equation:**
\[ A^2 - A = -I \]

3. **Factor out \( A \) from the left-hand side:**
\[ A(A - I) = -I \]

4. **Multiply both sides by \( A^{-1} \) (the inverse of \( A \)) from the left:**
\[ A^{-1} A (A - I) = A^{-1} (-I) \]
This simplifies to:
\[ (A - I) = -A^{-1} \]

5. **Rearranging gives us the expression for \( A^{-1} \):**
\[ A^{-1} = I - A \]

Thus, the inverse of \( A \) is:
\[ A^{-1} = I - A \]

### Final Answer:
The inverse of \( A \) is \( I - A \).