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We are tasked with finding the inverse of the given matrix \( A \) and showing that \( A^{-1} = \frac{A^2 - 3I}{2} \).

The matrix \( A \) is:

\[
A = \begin{bmatrix} 
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0 
\end{bmatrix}
\]

### Step 1: Compute \( A^2 \)
We first need to calculate the square of matrix \( A \). This is done by performing matrix multiplication: 

\[
A^2 = A \cdot A
\]

### Step 2: Compute \( 3I \)
The identity matrix \( I \) for a 3x3 matrix is:

\[
I = \begin{bmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\]

Then, \( 3I \) will be:

\[
3I = 3 \cdot I = \begin{bmatrix} 
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3 
\end{bmatrix}
\]

### Step 3: Verify \( A^{-1} = \frac{A^2 - 3I}{2} \)
Once we have computed \( A^2 \) and \( 3I \), we can subtract \( 3I \) from \( A^2 \), divide by 2, and verify that this gives the inverse \( A^{-1} \).

I'll now proceed to perform these computations to find the result.### Results:
1. The square of matrix \( A \) is:
   \[
   A^2 = \begin{bmatrix} 
   2 & 1 & 1 \\
   1 & 2 & 1 \\
   1 & 1 & 2 
   \end{bmatrix}
   \]

2. The expression \( \frac{A^2 - 3I}{2} \) yields:
   \[
   \frac{A^2 - 3I}{2} = \begin{bmatrix} 
   -0.5 & 0.5 & 0.5 \\
    0.5 & -0.5 & 0.5 \\
    0.5 & 0.5 & -0.5 
   \end{bmatrix}
   \]

3. The actual inverse of \( A \), \( A^{-1} \), is:
   \[
   A^{-1} = \begin{bmatrix} 
   -0.5 & 0.5 & 0.5 \\
    0.5 & -0.5 & 0.5 \\
    0.5 & 0.5 & -0.5 
   \end{bmatrix}
   \]

### Conclusion:
We have verified that \( A^{-1} = \frac{A^2 - 3I}{2} \), as both expressions yield the same result.

---

Would you like further details or clarifications?

Here are some related questions:
1. How can you find the inverse of a matrix using cofactor expansion?
2. What properties of symmetric matrices simplify finding their inverses?
3. How do elementary row operations relate to finding matrix inverses?
4. How would you compute the determinant of matrix \( A \)?
5. How does matrix inversion relate to solving systems of linear equations?

**Tip**: When performing matrix multiplication, ensure that the dimensions of the matrices match for valid operations.

Find A^(-1) if and show that A^(-1) = (A^2 - 3I) / 2 for the matrix A = [0 1 1; 1 0 1; 1 1 0].

Explore how to find the inverse of matrix A and prove that A^(-1) = (A^2 - 3I) / 2. Detailed steps include matrix multiplication, matrix identity properties, and linear algebra techniques.

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