https://file.aiquickdraw.com/mathbot/file/1727407985596r273o8do.jpg

The system of linear equations provided in the image is:

\[
\begin{aligned}
x + 2y &= -2 \\
y - z &= 0 \\
x + 3y - 2z &= 1
\end{aligned}
\]

We need to solve this using matrix inversion.

### Step 1: Write the system in matrix form

The system of equations can be written as:

\[
A \cdot \mathbf{x} = \mathbf{b}
\]

Where:
- \( A \) is the coefficient matrix
- \( \mathbf{x} \) is the vector of variables \( [x, y, z] \)
- \( \mathbf{b} \) is the constant matrix

For this system, the coefficient matrix \( A \) and the constant matrix \( \mathbf{b} \) are:

\[
A = \begin{pmatrix}
1 & 2 & 0 \\
0 & 1 & -1 \\
1 & 3 & -2
\end{pmatrix},
\quad
\mathbf{b} = \begin{pmatrix}
-2 \\
0 \\
1
\end{pmatrix}
\]

### Step 2: Find the inverse of matrix \( A \)

We can compute the inverse of \( A \) (denoted as \( A^{-1} \)) using standard methods such as the adjoint formula or row reduction, but for now, let me calculate the inverse directly:

\[
A^{-1} = \text{(inverse of A)}
\]

### Step 3: Multiply \( A^{-1} \) by \( \mathbf{b} \)

Once we have the inverse of \( A \), we can find the solution vector \( \mathbf{x} \) by multiplying \( A^{-1} \) by \( \mathbf{b} \):

\[
\mathbf{x} = A^{-1} \cdot \mathbf{b}
\]

Let me calculate the inverse of \( A \) and the solution for \( \mathbf{x} \).The solution to the system of linear equations using matrix inversion is:

\[
(x, y, z) = (4, -3, -3)
\]

Would you like further details on any of these steps?

Here are some related questions to explore this topic further:
1. How do you calculate the inverse of a matrix manually?
2. What conditions make a matrix invertible?
3. Can matrix inversion be used for all systems of linear equations?
4. How does matrix inversion differ from Gaussian elimination?
5. What are some real-world applications of solving systems of equations using matrix methods?

**Tip**: When dealing with large matrices, it's often more efficient to use methods like LU decomposition or other factorizations instead of directly computing the inverse!

Use matrix inversion to solve the given system of linear equations: x + 2y = -2, y - z = 0, x + 3y - 2z = 1.

Learn how to solve a system of linear equations using matrix inversion. This problem involves key concepts from linear algebra, such as matrix multiplication and finding the inverse of a matrix. Ideal for undergraduate or advanced high school students.

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