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The problem asks to find the inverse of the following \(3 \times 3\) matrix:

\[
A = \begin{pmatrix} 
1 & 1 & 1 \\ 
2 & -1 & 1 \\ 
3 & -2 & -1 
\end{pmatrix}
\]

### Steps to find the inverse of a \(3 \times 3\) matrix:

The inverse of a matrix \(A\), if it exists, can be found using the formula:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]

Where:
- \(\text{det}(A)\) is the determinant of matrix \(A\),
- \(\text{adj}(A)\) is the adjugate (transpose of the cofactor matrix) of \(A\).

#### 1. Find the determinant of the matrix \(A\):
\[
\text{det}(A) = 1 \left((-1)(-1) - 1(-2)\right) - 1 \left(2(-1) - 1(3)\right) + 1 \left(2(-2) - (-1)(3)\right)
\]

Let me calculate the determinant first, and then proceed to find the inverse.The determinant of the matrix \( A \) is \( \text{det}(A) = 7 \).

Since the determinant is non-zero, the matrix is invertible. Now, we will proceed to find the adjugate matrix and calculate the inverse of \( A \).

Let me compute the inverse of the matrix next.The inverse of the matrix \( A \) is:

\[
A^{-1} = \begin{pmatrix} 
0.4286 & -0.1429 & 0.2857 \\ 
0.7143 & -0.5714 & 0.1429 \\ 
-0.1429 & 0.7143 & -0.4286 
\end{pmatrix}
\]

If you need more details or have any questions, feel free to ask!

Here are 5 related questions:
1. How is the adjugate of a matrix calculated?
2. Why is the determinant important in determining the invertibility of a matrix?
3. What happens if the determinant of a matrix is zero?
4. Can all matrices be inverted? Why or why not?
5. How can you use matrix inverses to solve systems of equations?

**Tip**: The inverse of a matrix can also be used to solve linear systems using \( A^{-1} \cdot \mathbf{b} = \mathbf{x} \).

Dapatkan invers matriks berikut (3x3): [[1, 1, 1], [2, -1, 1], [3, -2, -1]]

Learn how to find the inverse of a 3x3 matrix [[1, 1, 1], [2, -1, 1], [3, -2, -1]]. This problem covers key linear algebra concepts such as matrix inversion, determinants, and adjugates. Step-by-step solutions provided. Suitable for Grades 11-12.

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