I have to solve the following matrix equation using Gauss Seidel :
$\begin{bmatrix}25 & 5 & 1\\64 & 8 & 1\\144&12&1 \end{bmatrix}$$\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}$= $\begin{bmatrix}106.8\\177.2\\279.2 \end{bmatrix}$
The coefficient matrix isn't diagonally dominant and it cannot be transformed into one. The solution doesn't converge. Even using under-relaxation doesn't help. Does this mean that iterative procedures cannot be used to solve this matrix equation?
From Wikipedia: the convergence properties of the Gauss–Seidel method are dependent on the matrix $A$. Namely, the procedure is known to converge if either:
The Gauss–Seidel method sometimes converges even if these conditions are not satisfied.
If we do the best we can with putting the matrix in DD form (the original matrix fails to converge), we have
$$\begin{bmatrix}144&12&1\\64 & 8 & 1\\25 & 5 & 1 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}= \begin{bmatrix}279.2\\177.2\\106.8 \end{bmatrix}$$
Starting from an initial point of $(1,1,0)$, after $140$ iterations, we converge to:
$$\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}= \begin{bmatrix}0.290476\\19.6905\\1.08572 \end{bmatrix}$$
Using a different solver, we get
$$\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}=\begin{bmatrix} 0.2904761904761902 \\ 19.6904761904762 \\ 1.085714285714253\\ \end{bmatrix}$$
Note: you can also use Gaussian Elimination on the original matrix to verify this result.
As to your original question, sometimes we can use these iterative solvers even when we cannot satisfy all of the conditions listed earlier.