Gaussianeliminációt

Gaussian elimination, also known as row reduction, is a fundamental algorithm in linear algebra used to solve systems of linear equations, find the rank of a matrix, and compute the inverse of a matrix. The process involves transforming a given matrix into an equivalent matrix in row echelon form or reduced row echelon form through a sequence of elementary row operations.

These 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 eliminate variables from the equations by creating zeros in specific positions within the matrix.

When applied to a system of linear equations represented by an augmented matrix, Gaussian elimination transforms the system into an equivalent one that is much easier to solve. This is typically achieved by obtaining a matrix in row echelon form, where the first non-zero element in each row (the pivot) is to the right of the pivot in the row above it, and all entries below the pivots are zero. Back-substitution can then be used to find the values of the variables.

Further reducing the matrix to reduced row echelon form, where all pivots are 1 and all entries above and below the pivots are zero, directly yields the solution to the system of equations without the need for back-substitution. Gaussian elimination is a computationally efficient and widely used method in various scientific and engineering disciplines.