• 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 represent 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 following
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. Interchange any two rows .

2. Replace any row by a nonzero 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.