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To solve this system of linear equations using the matrix inverse method, we start by rewriting the given equations in matrix form:

The system of equations is:
\[
3x_1 - x_2 - x_3 = 8
\]
\[
-x_1 + 3x_2 - x_3 = 16
\]
\[
-x_1 - x_2 + 3x_3 = 32
\]

This can be written in the matrix form \( AX = B \), where:

\[
A = \begin{bmatrix} 3 & -1 & -1 \\ -1 & 3 & -1 \\ -1 & -1 & 3 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad B = \begin{bmatrix} 8 \\ 16 \\ 32 \end{bmatrix}
\]

To find \( X \), we use the formula:
\[
X = A^{-1} B
\]
where \( A^{-1} \) is the inverse of matrix \( A \).

Let's calculate \( A^{-1} \) and then multiply it by \( B \) to find \( X \). I will perform these calculations now.The solution to the system of equations is:
\[
x_1 = 16, \quad x_2 = 18, \quad x_3 = 22
\]

Would you like more details on how the inverse matrix was calculated or any other aspect of the solution?

Here are five related questions to expand on this topic:
1. What are the conditions for a matrix to be invertible?
2. How can you solve a system of linear equations using other methods, such as Gaussian elimination?
3. What are the properties of an inverse matrix?
4. How do you find the determinant of a matrix, and why is it important for finding the inverse?
5. Can a non-square matrix have an inverse?

**Tip:** When solving linear systems using matrices, always ensure that the determinant of the matrix is non-zero, as a zero determinant indicates that the matrix does not have an inverse.

Tentukan penyelesaian SPL di bawah ini dengan menggunakan metode invers matrik:
3x1 - x2 - x3 = 8
-x1 + 3x2 - x3 = 16
-x1 - x2 + 3x3 = 32

Learn how to solve a system of linear equations using the matrix inverse method. This problem involves finding the solution to the equations 3x1 - x2 - x3 = 8, -x1 + 3x2 - x3 = 16, and -x1 - x2 + 3x3 = 32 by calculating the inverse of a matrix. Ideal for students studying linear algebra.

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