If `A^(2)-A+I=O`, then `A^(-1)` is equal to

Options

A. `A+I`
B. `A-I`
C. `A+2I`
D. `I-A`

Answer

To solve the equation \( A^2 - A + I = O \) and find \( A^{-1} \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ A^2 - A + I = O \]
This can be rearranged to:
\[ A^2 - A = -I \]

### Step 2: Multiply by \( A^{-1} \)
Next, we multiply both sides of the equation by \( A^{-1} \):
\[ A^2 A^{-1} - A A^{-1} = -I A^{-1} \]

### Step 3: Simplifying the Left Side
Using the property \( A A^{-1} = I \), we simplify the left side:
\[ A A^{-1} = I \implies A^2 A^{-1} = A \quad \text{and} \quad A A^{-1} = I \]
Thus, we have:
\[ A - I = -A^{-1} \]

### Step 4: Rearranging to Find \( A^{-1} \)
Now, we can rearrange this equation to isolate \( A^{-1} \):
\[ A^{-1} = I - A \]

### Conclusion
Therefore, the inverse of matrix \( A \) is:
\[ A^{-1} = I - A \]

### Final Answer
The correct option is \( I - A \).