If `A^(2) - A + I = 0 ` , then the inverse of the matrix A is

Options

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

Answer

To find the inverse of the matrix \( A \) given the equation \( A^2 - A + I = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ A^2 - A + I = 0 \]
Rearranging this, we get:
\[ A^2 - A = -I \]

### Step 2: Factoring the Left Side
We can factor the left side:
\[ A(A - I) = -I \]

### Step 3: Multiplying by the Inverse
To isolate \( A \), we can multiply both sides by \( A^{-1} \) (assuming \( A \) is invertible):
\[ A^{-1} A(A - I) = A^{-1}(-I) \]
This simplifies to:
\[ A - I = -A^{-1} \]

### Step 4: Rearranging for \( A^{-1} \)
Now, rearranging gives us:
\[ A^{-1} = I - A \]

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