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The image shows a system of linear equations that you are asked to solve using the matrix method. The system of equations is:

\[
\begin{aligned}
3x + 2y - z &= 14 \quad \text{(Equation 1)} \\
x - y + 2z &= -8 \quad \text{(Equation 2)} \\
2x + 3y - z &= 13 \quad \text{(Equation 3)}
\end{aligned}
\]

### Matrix Representation
This system can be written in matrix form as \( AX = B \), where:
\[
A = \begin{pmatrix}
3 & 2 & -1 \\
1 & -1 & 2 \\
2 & 3 & -1
\end{pmatrix},
\quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix},
\quad
B = \begin{pmatrix}
14 \\
-8 \\
13
\end{pmatrix}
\]

### Step 1: Find the Inverse of Matrix \( A \) (if it exists)
We first need to find the inverse of matrix \( A \), denoted as \( A^{-1} \), if it exists. The inverse of a 3x3 matrix \( A \) can be found using the formula:
\[
A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
\]
where:
- \(\text{det}(A)\) is the determinant of \( A \).
- \(\text{adj}(A)\) is the adjugate (or adjoint) of \( A \).

### Step 2: Compute the Determinant of \( A \)
The determinant of \( A \) is given by:
\[
\text{det}(A) = 3\left((-1)(-1) - 2(3)\right) - 2\left((1)(-1) - 2(2)\right) + (-1)\left((1)(3) - (-1)(2)\right)
\]

### Step 3: Compute the Adjugate of \( A \)
The adjugate of \( A \) is the transpose of the cofactor matrix.

### Step 4: Solve for \( X \)
Finally, once \( A^{-1} \) is computed, we can find \( X \) by multiplying \( A^{-1} \) by \( B \):
\[
X = A^{-1}B
\]

Let me calculate the inverse and solve for \( X \).### Solution

1. **Determinant of Matrix \( A \)**:  
   The determinant of \( A \) is \(\text{det}(A) = -10\), which is non-zero, so the inverse of \( A \) exists.

2. **Inverse of Matrix \( A \)**:  
   The inverse of \( A \) is:
   \[
   A^{-1} = \begin{pmatrix}
   0.5 & 0.1 & -0.3 \\
   -0.5 & 0.1 & 0.7 \\
   -0.5 & 0.5 & 0.5
   \end{pmatrix}
   \]

3. **Solution Vector \( X \)**:
   Multiplying \( A^{-1} \) by \( B \) gives the solution:
   \[
   X = \begin{pmatrix}
   x \\
   y \\
   z
   \end{pmatrix}
   = \begin{pmatrix}
   2.3 \\
   1.3 \\
   -4.5
   \end{pmatrix}
   \]

### Final Answer
- \( x = 2.3 \)
- \( y = 1.3 \)
- \( z = -4.5 \)

Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. What are the conditions for a matrix to be invertible?
2. How can matrix multiplication be used to verify solutions?
3. What are some alternative methods to solve a system of linear equations?
4. Can the matrix method be applied to non-square matrices?
5. What is the significance of the determinant in linear algebra?

**Tip:** Always check the determinant of a matrix before attempting to find its inverse. If the determinant is zero, the matrix is singular and does not have an inverse.

Use the matrix method to solve the system of linear equations: 3x + 2y - z = 14, x - y + 2z = -8, 2x + 3y - z = 13.

Learn how to solve a system of linear equations using the matrix method. This example covers the matrix equation AX = B, finding the inverse of a matrix, and calculating the determinant for a 3x3 matrix. Suitable for students in Grades 10-12.

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