If A is square matrix such that |A| `ne` 0 and `A^2-A+2I=O " then " A^(-1)`=?

Options

A. (I-A)
B. (I+A)
C. `1/2(I-A)`
D. `I/2(I+A)`

Explanation

To solve the problem, we need to find the inverse of the matrix \( A \) given the equation:

\[ A^2 - A + 2I = 0 \]

### Step 1: Rearranging the Equation
First, we rearrange the given equation to isolate \( A^2 \):

\[ A^2 = A - 2I \]

**Hint:** Rearranging equations can help simplify the problem and isolate the terms you need.

### Step 2: Multiplying by \( A^{-1} \)
Next, we multiply both sides of the equation by \( A^{-1} \):

\[ A^{-1} A^2 = A^{-1}(A - 2I) \]

This simplifies to:

\[ A A = I A - 2 A^{-1} I \]

**Hint:** Remember that \( A^{-1} A = I \) (the identity matrix), which simplifies the left-hand side.

### Step 3: Simplifying the Left Side
The left side simplifies to:

\[ A = A - 2A^{-1} \]

**Hint:** Keep track of your terms; simplifying both sides can help you see relationships between them.

### Step 4: Isolating \( A^{-1} \)
Now, we can isolate \( A^{-1} \):

\[ A + 2A^{-1} = A \]

Subtract \( A \) from both sides:

\[ 2A^{-1} = I - A \]

**Hint:** Isolating variables is key to solving equations. Here, we are isolating \( A^{-1} \).

### Step 5: Dividing by 2
Now, divide both sides by 2 to solve for \( A^{-1} \):

\[ A^{-1} = \frac{1}{2}(I - A) \]

**Hint:** When you have a coefficient in front of a variable, dividing by that coefficient can help isolate the variable.

### Final Answer
Thus, the inverse of the matrix \( A \) is:

\[ A^{-1} = \frac{1}{2}(I - A) \]

### Summary of Steps
1. Rearranged the original equation.
2. Multiplied by \( A^{-1} \).
3. Simplified the left side.
4. Isolated \( A^{-1} \).
5. Divided by 2 to find \( A^{-1} \).