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}

$$
B =
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 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.

Contents

Examples

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 & rac{1}{5} \
0 & 1 & -rac{1}{3} & rac{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:
$$
egin{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}

the augmented matrix would be composed of
$$
A =
egin{bmatrix}
1 & 2 & 3 \
3 & 4 & 7 \
6 & 5 & 9
end{bmatrix}

and
$$
B =
egin{bmatrix}
0 \
2 \
11
end{bmatrix}

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