04 Mar 2026

Gauss Elimination Method

Solving linear simultaneous equations by forward elimination and back substitution.

msc semester-i numerical-methods linear-equations

Gauss elimination is a direct method for solving a system of linear equations:

\[A\mathbf{x}=\mathbf{b}.\]

The method converts the coefficient matrix into upper triangular form and then solves the unknowns by back substitution.

Augmented matrix

For three equations, write

\[\left[ \begin{array}{ccc|c} a_{11} & a_{12} & a_{13} & b_1\\ a_{21} & a_{22} & a_{23} & b_2\\ a_{31} & a_{32} & a_{33} & b_3 \end{array} \right].\]

Row operations are then used to remove the coefficients below the diagonal.

Forward elimination

The goal is to obtain

\[\left[ \begin{array}{ccc|c} u_{11} & u_{12} & u_{13} & c_1\\ 0 & u_{22} & u_{23} & c_2\\ 0 & 0 & u_{33} & c_3 \end{array} \right].\]

This corresponds to the triangular system

\[\begin{aligned} u_{11}x+u_{12}y+u_{13}z&=c_1,\\ u_{22}y+u_{23}z&=c_2,\\ u_{33}z&=c_3. \end{aligned}\]

Back substitution

First find

\[z=\frac{c_3}{u_{33}}.\]

Then substitute $z$ into the second equation to find $y$, and finally substitute $y$ and $z$ into the first equation to find $x$.

Pivoting

If a pivot element is zero or very small, rows should be interchanged. This improves numerical stability.

Key points

• Gauss elimination is efficient for small and medium linear systems.
• It is sensitive to round-off error when pivots are small.
• Partial pivoting is usually safer than plain elimination.