Apply the matrix method to solve a system of three linear equations, such as: $\left\{\begin{array}{l} 3x + 8y + 2z = -5 \\ 2x + 5y - 3z = 0 \\ x + 2y - 2z = -1 \end{array}\right.$ First, write the augmented matrix for the system. Next, systematically apply row operations to achieve row-echelon form, ensuring the diagonal entries are $1$ and the entries below them in their respective columns are $0$. Once the matrix is in row-echelon form, write the corresponding system of equations. Finally, use substitution to find the remaining variables, and write the solution as an ordered triple. Always check that the solution makes the original equations true.