Augmented matrix

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

Given the matrices "A" and "B", where:

$$A = egin{bmatrix} 1 & 3 & 2 \ 2 & 0 & 1 \ 5 & 2 & 2 end{bmatrix}, quadB = egin{bmatrix} 4 \ 3 \ 1 end{bmatrix}

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

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

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.

Examples

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

$$(I|C^{-1}) = egin{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 and the solution vector of each equation set.For the set of equations:

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

the augmented matrix would be composed of

$$A =egin{bmatrix}1 & 2 & 3 \3 & 4 & 7 \6 & 5 & 9end{bmatrix}, quadB = egin{bmatrix}0 \2 \11end{bmatrix}

Leaving us with:

$$C =egin{bmatrix}1 & 2 & 3 & 0 \3 & 4 & 7 & 2 \6 & 5 & 9 & 11end{bmatrix}.

References

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