Consider the following set of equations. For the Gauss-Seidel method, will this set of equations converge to a solution?
3x + 6y + 2z = 9
12x + 7y + 3z = 17
2x + 7y - 11z = 49
Since the matrix is not diagonally dominant, it will not converge to the correct answer. This set will not converge to the correct answer.
The matrix is non-orthogonal; hence, it will not converge.
There is no way to really tell unless you try the method and plot the results.