Augmented Matrices for Systems

Each system of linear equations has a corresponding augmented matrix.

Example 1

Write the system below as an augmented matrix.

( i ) 3x + 2y = 6

( ii ) -x + 5y = 8

Solution .

The augmented matrix for this system is given below.

Note that row one of the matrix contains the coefficients of x and y in the first equation along with the constant 6 that is on the right side of equation one. In a similar way, the values in the second row of the matrix come from the second equation of the system.

A matrix is just a rectangular array of numbers. The term augmented is used because the coefficients of the variables as well as the constants on the right side of the equals sign are included. In some texts you will see a dotted vertical line before the last column to show where the equals sign was.

The techniques for solving linear systems can easily be adapted to this matrix form and carried out by computers, calculators, and even people using paper and pencils.

Example 2

Convert the augmented matrix given below into a linear system with variables x and y.

Solution .

The system corresponding to this is

( i ) 3x - 2y = 12

( ii ) y = 5

Example 3

Convert the augmented matrix which appear below into a linear system with variables x and y.

Solution .

The linear system is

( i ) 12x + 7y = 45

( ii ) -x + 3y = 13

( iii ) 9x + 3y = 28

Example 4

Convert the linear system given below into an augmented matrix

( i ) 2x + 8y - 3z = 20

( ii ) x + 9z = -23

( iii) 3y = 5

Solution .