Augmented matrix

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In linear algebra, the augmented matrix of a matrix is obtained by combining two 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:

$(A|B)= \begin{bmatrix} 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end{bmatrix}$

This is useful when solving systems of linear equations given by square matrices. They may also be used to find the inverse of a matrix. By reducing the matrix into row-echelon form, where the consistency (or inconsistency) of the system can be read off.

[edit] Examples

Let C be a square 2×2 matrix where $C = \begin{bmatrix} 1 & 3 \\ -5 & 0 \end{bmatrix}$

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).

$(C|I) = \begin{bmatrix} 1 & 3 & 1 & 0\\ -5 & 0 & 0 & 1 \end{bmatrix}$

$(I|C^{-1}) = \begin{bmatrix} 1 & 0 & 0 & -\frac{1}{5} \\ 0 & 1 & \frac{1}{3} & \frac{1}{15} \end{bmatrix}$

As used in linear algebra, an augmented matrix is used to represent the coefficients as well as the constants of each equation. For the set of equations:

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