Augmented matrix
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In linear algebra, the augmented matrix of a matrix 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:
![(A|B)=
\left[\begin{array}{ccc|c}
1 & 3 & 2 & 4 \\
2 & 0 & 1 & 3 \\
5 & 2 & 2 & 1
\end{array}\right]](http://upload.wikimedia.org/math/8/c/b/8cbfd0eec007aa6285955b8ed1a9de8b.png)
This is useful when solving systems of linear equations or the augmented matrix may also be used to find the inverse of a matrix by combining it with the identity matrix.
[edit] 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. We then reduce the part of (C|I) corresponding to C to the identity matrix using only elementary matrix transformations on (C|I).
![(C|I) =
\left[\begin{array}{cc|cc}
1 & 3 & 1 & 0\\
-5 & 0 & 0 & 1
\end{array}\right]](http://upload.wikimedia.org/math/2/2/8/2287af9b77a63d0c27d9d220fda56960.png)
![(I|C^{-1}) =
\left[\begin{array}{cc|cc}
1 & 0 & 0 & -\frac{1}{5} \\
0 & 1 & \frac{1}{3} & \frac{1}{15}
\end{array}\right]](http://upload.wikimedia.org/math/7/2/c/72cb9fd6adf4343b3c37567082e59ddf.png)
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:
![(A|B) =
\left[\begin{array}{ccc|c}
1 & 2 & 3 & 0 \\
3 & 4 & 7 & 2 \\
6 & 5 & 9 & 11
\end{array}\right]](http://upload.wikimedia.org/math/a/8/3/a839cf557107100783576aa71de34241.png)
. ,or
![(A|B) =
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 4 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & -2 \\
\end{array}\right]](http://upload.wikimedia.org/math/2/c/4/2c414493b8f3e9456c637459cbc79f33.png)
.
[edit] References
- Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, ISBN 0-486-67102-X. Page 31.
