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HomeHomework HelpmathematicsElementary Matrices

Elementary Matrices

Elementary matrices are matrices that represent elementary row operations on other matrices, and they play a crucial role in linear algebra, particularly in the context of matrix factorization and solving linear systems.

intermediate
2 hours
Mathematics
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Overview

Elementary matrices play a vital role in linear algebra by simplifying matrix operations and providing a systematic approach to solving linear systems. They are derived from the identity matrix through elementary row operations, which include row swapping, scaling, and addition. Understanding these ...

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Key Terms

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

Example: A 2x3 matrix has 2 rows and 3 columns.

Identity Matrix
A square matrix with ones on the diagonal and zeros elsewhere.

Example: The 2x2 identity matrix is [[1, 0], [0, 1]].

Elementary Row Operation
An operation that can be performed on the rows of a matrix to simplify it.

Example: Swapping two rows is an elementary row operation.

Inverse Matrix
A matrix that, when multiplied by the original matrix, yields the identity matrix.

Example: The inverse of [[1, 2], [3, 4]] is [[-2, 1], [1.5, -0.5]].

Row Echelon Form
A form of a matrix where all non-zero rows are above any rows of all zeros.

Example: A matrix in row echelon form has leading 1s in each row.

Linear System
A collection of linear equations that can be solved simultaneously.

Example: The equations x + y = 2 and x - y = 0 form a linear system.

Related Topics

Matrix Operations
Study the various operations that can be performed on matrices, including addition, subtraction, and multiplication.
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Linear Transformations
Explore how matrices can represent linear transformations in vector spaces.
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Determinants and Their Properties
Learn about determinants, their calculation, and their significance in linear algebra.
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Key Concepts

Elementary Row OperationsTypes of Elementary MatricesMatrix InversionApplications in Linear Algebra