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To find the inverse of the matrix \( A \) given by:

\[
A = \begin{bmatrix} -1 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{bmatrix}
\]

we need to compute \( A^{-1} \), which requires calculating the matrix's determinant and its adjugate.

### Step-by-Step Solution:
1. **Calculate the Determinant** of \( A \):
   \[
   \det(A) = -1 \begin{vmatrix} -2 & 1 \\ 1 & -2 \end{vmatrix} - 1 \begin{vmatrix} 1 & 1 \\ 1 & -2 \end{vmatrix} + 1 \begin{vmatrix} 1 & -2 \\ 1 & 1 \end{vmatrix}
   \]
   Each minor determinant can be calculated separately.

2. **Find the Matrix of Cofactors** by determining the cofactors of each element in \( A \).

3. **Transpose the Matrix of Cofactors** to get the adjugate matrix.

4. **Multiply the Adjugate by \( \frac{1}{\det(A)} \)** to get \( A^{-1} \).

Let me know if you would like a detailed walkthrough of each step, or if you want to calculate a specific part together.

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### Related Questions:
1. How do you calculate the determinant of a 3x3 matrix?
2. What is the formula for the adjugate of a matrix?
3. Why do we transpose the cofactor matrix to find the adjugate?
4. Can every square matrix be inverted?
5. How can we verify if a calculated inverse matrix is correct?

### Tip:
When calculating determinants, simplifying each minor step-by-step can help avoid common arithmetic errors.

If A = [[-1, 1, 1], [1, -2, 1], [1, 1, -2]], find the inverse of matrix A, denoted as A^{-1}.

Learn how to find the inverse of a 3x3 matrix A = [[-1, 1, 1], [1, -2, 1], [1, 1, -2]]. This problem covers matrix inversion, including the calculation of determinants, cofactors, and the adjugate matrix. Suitable for undergraduate students studying linear algebra.

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