Augmented matrix

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:

$A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & 0 & 1 \\ 5 & 2 & 2 \end{bmatrix} , \quad B = \begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix}.$

Then, the augmented matrix (A|B) is written as:

$(A|B)= \left[\begin{array}{ccc|c} 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end{array}\right].$

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.

Examples

$(C|I) = \left[\begin{array}{cc|cc} 1 & 3 & 1 & 0\\ -5 & 0 & 0 & 1 \end{array}\right]$

$(I|C^{-1}) = \left[\begin{array}{cc|cc} 1 & 0 & 0 & -\frac{1}{5} \\ 0 & 1 & \frac{1}{3} & \frac{1}{15} \end{array}\right]$

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:

$\begin{align} x_1 + 2x_2 + 3x_3 &= 0 \\ 3x_1 + 4x_2 + 7x_3 &= 2 \\ 6x_1 + 5x_2 + 9x_3 &= 11 \end{align}$

the coefficients and constant terms give the matrices

$A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 7 \\ 6 & 5 & 9 \end{bmatrix} , \quad B = \begin{bmatrix} 0 \\ 2 \\ 11 \end{bmatrix}.$

Giving the augmented matrix:

$(A|B) = \left[\begin{array}{ccc|c} 1 & 2 & 3 & 0 \\ 3 & 4 & 7 & 2 \\ 6 & 5 & 9 & 11 \end{array}\right]$

References

• Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, ISBN 0-486-67102-X. Page 31.