• 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.
horizontally, columns run vertically.
• Examples :
• An augmented matrix can be used to represent a system of
The system is represented as
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.
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
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
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
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
The solution is x = 1, y = 0, z = 2 .
• Example of an inconsistent system.
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.)
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