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