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The problem asks to find the inverse of matrix \( A = \begin{pmatrix} 1 & 1 & -1 \\ 1 & 3 & 1 \\ 2 & 2 & -1 \end{pmatrix} \).

### To find the inverse of a 3x3 matrix, follow these steps:

1. **Calculate the determinant of matrix \( A \)**:
   \[
   \text{det}(A) = 1 \cdot \begin{vmatrix} 3 & 1 \\ 2 & -1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} + (-1) \cdot \begin{vmatrix} 1 & 3 \\ 2 & 2 \end{vmatrix}
   \]
   We will evaluate each of these 2x2 determinants.

2. **Cofactor expansion and adjugate matrix**:
   After finding the determinant, we compute the cofactors for each element of matrix \( A \) to construct the adjugate matrix.

3. **Divide adjugate matrix by the determinant**:
   If the determinant is non-zero, the inverse is given by:
   \[
   A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A)
   \]

Let me compute the detailed solution for you now.The determinant of matrix \( A \) is \( \text{det}(A) = 2.0 \), which is non-zero, so the inverse exists.

The inverse of matrix \( A \) is:
\[
A^{-1} = \begin{pmatrix}
-2.5 & -0.5 & 2 \\
1.5 & 0.5 & -1 \\
-2 & 0 & 1
\end{pmatrix}
\]

Would you like more details on how the inverse was calculated, or do you have any other questions?

Here are 5 related questions to expand on this:

1. How is the determinant of a matrix calculated?
2. What does it mean when the determinant of a matrix is zero?
3. How is the adjugate matrix related to the inverse?
4. Can every square matrix have an inverse?
5. What are the properties of an inverse matrix?

**Tip**: The inverse of a matrix is used in solving systems of linear equations, especially when dealing with matrix equations of the form \( A\mathbf{x} = \mathbf{b} \).

Tìm ma trận nghịch đảo của ma trận A = \begin{pmatrix} 1 & 1 & -1 \\ 1 & 3 & 1 \\ 2 & 2 & -1 \end{pmatrix}

Learn how to find the inverse of a 3x3 matrix A = \begin{pmatrix} 1 & 1 & -1 \\ 1 & 3 & 1 \\ 2 & 2 & -1 \end{pmatrix using key matrix algebra concepts such as the determinant, cofactor expansion, and adjugate matrix. Suitable for advanced high school or college students.

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