math111_logo Augmented Matrix


In systems of linear equations, the notations for the variables are not essential.

Example The essential information of the following system

3x1 + x2 - x3 = 2
x1 - x2 + x3 = 2
2x1 + 2x2 + x3 = 6

is contained in the matrix

[ 3 1 -1 2 ]
1 -1 1 2
2 2 1 6

made up of the coefficients on the left side and the numbers on the right side. Conversely, the system can be recovered from the matrix. For example, the following matrix

[ 1 -2 0 3 ]
0 1 -1 -2
-2 0 -5 1

comes from the following system (provided x1, x2, x3 are chosen as the variables).

x1 - 2x2 = 3
x2 - x3 = -2
- 2x1 - 5x3 = 1

In general, the essential information of a system of linear equations

a11x1 + a12x2 + ... + a1nxn = b1
a21x1 + a22x2 + ... + a2nxn = b2
... ... ...
am1x1 + am2x2 + ... + amnxn = bm

is contained in the augmented matrix

[A b] = [ a11 a12 ... a1n b1 ]
a21 a22 ... a2n b2
: : : :
am1 am2 ... amn bm

which is made up of the coefficient matrix

A = [ a11 a12 ... a1n ]
a21 a22 ... a2n
: : :
am1 am2 ... amn

and the right side vector.

b = [ b1 ]
b2
:
bm

We also denote the left side of the system by

Ax = [ a11x1 + a12x2 + ... + a1nxn ]
a21x1 + a22x2 + ... + a2nxn
:
am1x1 + am2x2 + ... + amnxn

so that the system of linear equations is Ax = b.

Example Consider the system

x + 2y = 3
4x + 5y = 6
7x + 8y = 9
10x + 11y = 12

The coefficient matrix is

A = [ 1 2 ]
4 5
7 8
10 11

the right side vector is

b = [ 3 ]
6
9
12

And the augmented matrix is

[A b] = [ 1 2 3 ]
4 5 6
7 8 9
10 11 12

Finally, we note the following correspondences between the augmented matrix and the system.

columns of A ⇔ variables of the system Ax = b
rows of [A b] ⇔ equations in the system Ax = b


[exercise]
[previous topic] [next topic]