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Augmented matrix

 

 
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Augmented matrix
 
 
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In linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, the augmented matrix of a matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 is obtained by changing a matrix in some way.

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
System of linear equations

In mathematics, a system of linear equations is a collection of linear equations involving the same set of variables. For example,is a system of three equations in the three variables ....
 or the 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-by-n square matrix with ones on the main diagonal and zeros elsewhere....
.

C be a square 2×2 matrix where

To find the inverse of C we create (C|I) where I is the 2×2 identity matrix
Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere....
.






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In linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, the augmented matrix of a matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 is obtained by changing a matrix in some way.

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
System of linear equations

In mathematics, a system of linear equations is a collection of linear equations involving the same set of variables. For example,is a system of three equations in the three variables ....
 or the 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-by-n square matrix with ones on the main diagonal and zeros elsewhere....
.

Examples

Let C be a square 2×2 matrix where

To find the inverse of C we create (C|I) where I is the 2×2 identity matrix
Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere....
. We then reduce the part of (C|I) corresponding to C to the identity matrix using only elementary matrix transformations on (C|I).


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 augmented matrix would be composed of

Leaving us with:

.