Apply the matrix method to determine if a system of linear equations is inconsistent, such as: $\left\{\begin{array}{l} x + y + 3z = 0 \\ x + 3y + 5z = 0 \\ 2x + 4z = 1 \end{array}\right.$ First, write the augmented matrix for the equations. Then, use row operations to systematically eliminate variables, working towards row-echelon form. If during this process a row emerges with all zeros on the left side of the vertical line and a non-zero number on the right, it translates to a mathematically false statement like $0 = 1$. Because this results in a false statement, it indicates the system is inconsistent and has no solution.