The solution to the system of linear equations using the matrix inversion method is:

$x = 6, \quad y = -2, \quad z = 4$

Would you like the detailed steps for solving this using Cramer's rule, Gaussian elimination, or Gauss-Jordan method?

Here are 5 follow-up questions that might interest you:

1. What is Cramer's rule, and how does it apply to this system of equations?
2. Can you explain the steps of Gaussian elimination in detail for this system?
3. How does the Gauss-Jordan method differ from regular Gaussian elimination?
4. What are the advantages and limitations of using matrix inversion to solve linear systems?
5. Could we solve this system graphically? If so, how?

Tip: Always check for the determinant of the matrix when using matrix inversion, as it must be non-zero for the system to have a unique solution.