22、Gaussian Elimination: A Comprehensive Guide

Gaussian Elimination: A Comprehensive Guide

1. Introduction to Gaussian Elimination

Gaussian elimination is a fundamental method for solving systems of linear equations. Consider a system of two equations:
[
\begin{cases}
a_{11}x_1 + a_{12}x_2 = b_1 \
a_{21}x_1 + a_{22}x_2 = b_2
\end{cases}
]
We can solve for (x_2) and (x_1) using the following formulas:
[x_2 = \frac{a_{11}b_2 - a_{21}b_1}{a_{11}a_{22} - a_{21}a_{12}}]
[x_1 = \frac{a_{22}b_1 - a_{12}b_2}{a_{11}a_{22} - a_{21}a_{12}}]
These formulas can also be derived from Cramer’s rule.

The elimination of unknowns can be extended to systems with more than two or three equations. However, for larger systems, the calculations become extremely tedious to