GaußEliminationsverfahren

Gauss Elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations, find the rank of a matrix, and calculate the inverse of an invertible square matrix. The process involves transforming a given matrix into row echelon form or reduced row echelon form through a sequence of elementary row operations.

The elementary row operations are: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row. The goal is to systematically introduce zeros below the leading non-zero entry (pivot) in each row. This process is typically performed on an augmented matrix, which combines the coefficient matrix of the system with the constant vector.

Once the matrix is in row echelon form, the system of equations can be solved using back-substitution. If the matrix is further reduced to reduced row echelon form, the solution can be read directly from the augmented column. Gauss elimination is an effective method for solving systems with any number of variables, though its computational complexity increases with the size of the system. It forms the basis for many other numerical algorithms and is a cornerstone of computational mathematics.