Augmented matrix
Encyclopedia
In linear algebra
, an augmented matrix is a matrix
obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations
on each of the given matrices.
Given the matrices A and B, where:
Then, the augmented matrix (A|B) is written as:
This is useful when solving systems of linear equations.
The number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the corresponding augmented matrix.
An augmented matrix may also be used to find the inverse of a matrix by combining it with the identity matrix
.
For the set of equations:
the coefficients and constant terms give the matrices
Giving the augmented matrix:
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, an augmented matrix is a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations
Elementary row operations
In mathematics, an elementary matrix is a simple matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group of invertible matrices...
on each of the given matrices.
Given the matrices A and B, where:
Then, the augmented matrix (A|B) is written as:
This is useful when solving systems of linear equations.
The number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the corresponding augmented matrix.
An augmented matrix may also be used to find the inverse of a matrix by combining it with the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
.
Examples
Solution of a linear system
As used in linear algebra, an augmented matrix is used to represent the coefficients and the solution vector of each equation set.For the set of equations:
the coefficients and constant terms give the matrices
Giving the augmented matrix: