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HomeHomework HelpmathematicsGaussian Elimination

Gaussian Elimination

A mathematical algorithm used to transform a matrix into row echelon form (REF) or reduced row echelon form (RREF) through a series of elementary row operations, which is essential for solving systems of linear equations and finding the rank of a matrix

intermediate
3 hours
Mathematics
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Overview

Gaussian elimination is a systematic method used to solve systems of linear equations. By transforming the augmented matrix into row echelon form through elementary row operations, it simplifies the process of finding solutions. This method is not only fundamental in mathematics but also has practic...

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Key Terms

Matrix
A rectangular array of numbers arranged in rows and columns.

Example: A 2x2 matrix looks like this: [[1, 2], [3, 4]].

Row Echelon Form
A form of a matrix where all non-zero rows are above any rows of all zeros, and leading coefficients are to the right.

Example: The matrix [[1, 2, 3], [0, 1, 4]] is in row echelon form.

Elementary Row Operations
Operations that can be performed on rows of a matrix: row swapping, scaling, and addition.

Example: Swapping two rows is an elementary row operation.

Back Substitution
A method used to solve for variables in a system of equations after obtaining row echelon form.

Example: In the equation x + 2y = 5, if y = 2, then x = 1.

Augmented Matrix
A matrix that includes the coefficients and constants of a system of equations.

Example: For the equations x + y = 2 and 2x + 3y = 5, the augmented matrix is [[1, 1, 2], [2, 3, 5]].

Leading Coefficient
The first non-zero number in a row of a matrix.

Example: In the row [0, 1, 2], the leading coefficient is 1.

Related Topics

Matrix Inversion
The process of finding a matrix that, when multiplied with the original matrix, yields the identity matrix.
intermediate
Linear Programming
A method for achieving the best outcome in a mathematical model whose requirements are represented by linear relationships.
advanced
Determinants
A scalar value that can be computed from the elements of a square matrix, providing important properties of the matrix.
intermediate

Key Concepts

Row Echelon FormElementary Row OperationsBack SubstitutionMatrix Representation