• A matrix is a rectangular array of numbers, enclosed in
brackets. The numbers are called the

entries of the matrix. Entries are identified by their row and column position.
Rows run

horizontally, columns run vertically.

• Examples :

• An augmented matrix can be used to re present a system of
equations.

The system is represented as

Notice that equation 1 becomes row 1, equation 2 becomes
row 2, the x terms are in column 1, the

y terms are in column 2, and the equal signs are represented by the vertical
line.

• Write the augmented matrix that represents the fol lowing
system of equations.

• Write the system of equations that corresponds to the
following augmented matrix.

• Write the system of equations that corresponds to the
following augmented matrix.

• To solve a system of equations using its augmented
matrix representation, we will transform the

original augmented matrix into a form similar to the previous example. This will
allow us to read

the solutions of the system .

• There are three row operations that can be applied to an
augmented matrix. These correspond to

algebraic operations that can be applied to the corresponding system of
equations.

• Row operations

1. Inter change any two rows .

2. Replace any row by a non zero constant multiple of that row .

3. Replace any row by the sum of that row and a constant multiple of another
row.

• Matrix method - an example.

Solve

Step 1 : Write the augmented matrix.

Step 2 : Use row operations to transform the augmented
matrix into the form

which has solutions x = a, y = b .

The last augmented matrix corresponds to the system

which has solution x = 1, y = 2.

Check :

5(1) + 10(2) = 25

10(1) + 12(2) = 34

• The strategy for transforming the original augmented
matrix using row operations:

1. Place a 1 in row1, column 1

2. Place 0's in all other entries in column 1 - leaving the 1 in row 1, column 1
unchanged

3. Place a 1 in row 2, column 2

4. Place 0's in all other entries in column 2 - leaving the 1 in row 2, column 2
unchanged

5. Continue this pattern. Place a 1 in row n, column n. Place 0's in all other
entries of column n -

leaving the 1 in row n, column n unchanged.

6. If a row is obtained that contains only 0's to the left of the vertical bar,
place it at the bottom of

the matrix.

A matrix generated using the strategy outlined above is
said to be in row-echelon form.

• Solve by writing the augmented matrix in row-echelon
form.

The solution is x = 1, y = 0, z = 2 .

• Example of an inconsistent system.

Solve

Notice that row 2 corresponds to the equation 0 = 31, a
contradiction. Therefore, this

system has no solution. It is inconsistent.

• Example of a consistent system with dependent equations.

Notice that row 2 corresponds to the equation 0 = 0, an
identity. This indicates that

equation 2 can be derived from equation 1. They are equivalent equations .

Any point on the line
is a solution of the system.

Solutions: where y is
any real number . ( y is called a parameter.)

Give three different solutions for this system.

• Use the matrix method to solve the following system.