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.