Gauss Elimination Method

The Gauss Elimination Method (GEM) is the method for solving simultaneous equations. The GEM consists of two parts and these are the elimination phase and the solution phase. The function of the elimination phase is to trasnform the equations into the form \(\textbf{Ux} = \textbf{c}\). These equations in the solution phase are solved using back substitution.

Example 1

Solve system of linear equations using GEM.

$$ 4x_1-2x_2+x_3 = 11\qquad (a)\\ -2x_1+4x_2-2x_3 = -16 \qquad (b)\\ x_1-2x_2+4x_3 = 17 \qquad (c)\\ $$

The Elimination Phase

The elimination phase multiplies one equation (equation j) by a constant \(\lambda\) and subtracting it from another equation (equqation i). This can be written in the following form: $$ Eq.(i) <- Eq. (i) - \lambda \times Eq. (j) $$ The Eq.(j) is called the pivot equation (equation being subtracted). The elimination process in Example 1 starts by taking the Eq.(a) to be the pivo equation and choosing the \(\lambda\) coefficinet value to eliminate \(x_1\) from Eqs.(b), and (c). $$ Eq.(b) <- Eq.(b)-(-0.5)\times Eq.(a)\qquad\\ Eq.(c) <- Eq.(c)-0.25\times Eq.(a) $$ By performing the previous procedure the equations can be written as: $$ 4x_1-2x_2+x_3 = 11\qquad (a)\\ 3x_2-1.5x_3 = -10.5 \qquad (b)\\ 1.5x_2+3.75x_3 = 14.25 \qquad (c)\\ $$ The next step is to use the Eq.(b) as the pivot equation and eliminate \(x_2\) from Eq.(c): $$Eq.(c) <- Eq.(c) - (-0.5) \times Eq.(b)$$ By performing the previous procedure the equations take the following form: $$ 4x_1-2x_2+x_3 = 11\qquad (a)\\ 3x_2-1.5x_3 = -10.5 \qquad (b)\\ 3x_3 = 9 \qquad (c)\\ $$ The elimination process is completed and the original equations were replaced by equivalent equations that can be solved using the back substitution.

The Back Substituion Phase

After the elimination process is completed the unknowns can be computed using back substitution. Solving Eqs.(c), (b), and (a) the following solutions are obtained: $$ x_3 = \frac{9}{3} = 3 \qquad \\ x_2 = \frac{-10.5 + 1.5x_3}{3} = \frac{-10.5 + 1.5(3)}{3} = -2 \\ x_1 = \frac{11+2x_2-x_3}{4} = \frac{11+2(-2)-3}{4} = 1 $$

Gauss Elimination for 3x3 matrix

\(a_{11}\) = \(a_{12}\) = \(a_{13}\) = \(x_{1}\) =
\(a_{21}\) = \(a_{22}\) = \(a_{23}\) = \(x_2\) =
\(a_{31}\) = \(a_{32}\) = \(a_{33}\) = \(x_3\) =
\(R_1\) = \(R_2\) = \(R_3\) =