2. Chapter 7: Systems of Equations and Inequalities; Matrices
7.2 Solution of Linear Systems in Three Variables
7.3 Solution of Linear Systems by Row Transformations
7.4 Matrix Properties and Operations
7.5 Determinants and Cramer’s Rule
7.6 Solution of Linear Systems by Matrix Inverses
7.7 Systems of Inequalities and Linear Programming
7.8 Partial Fractions

3. 7.3 Solution of Linear Systems by Row Transformations
Matrices and Technology
Matrix methods suitable for calculator and computer solutions of large systems
Matrix Row Transformations
Streamlined use of echelon methods

4. 7.3 Solution of Linear Systems by Row Transformations
This is called an augmented matrix where each member of the array is called an element or entry.
The rows of an augmented matrix can be treated just like the equations of a linear system.

5. 7.3 Matrix Transformations
For any augmented matrix of a system of linear
equations, the following row transformations will
result in the matrix of an equivalent system.
Any two rows may be interchanged.
The elements of any row may be multiplied by a nonzero real number.
Any row may be changed by adding to its elements a multiple of the corresponding elements of another row.

6. 7.3 Solving a System by the Row Echelon Method
Example Solve the system.
Analytic Solution The augmented matrix for the
system is

7. 7.3 Solving a System by the Row Echelon Method
The matrix represents the system
The solution to this system using back-substitution is
{(1, –2)}.

8. 7.3 Solving a System by the Row Echelon Method
Graphing Calculator Solution
Enter the augmented matrix of the system.
Using the ref command on the TI-83, we can find the
row echelon form of the matrix.

9. 7.3 Solving a System by the Row Echelon Method
The left screen displays decimal entries, but can be
converted to fractions as indicated in the right
screen. This corresponds to the augmented matrix in
the analytic solution. The rest of the solution process
is the same.

10. 7.3 Extending the Row Echelon Method to Larger Systems
Example Solve the system.
The augmented matrix of the system is

11. 7.3 Extending the Row Echelon Method to Larger Systems
There is already a 1 in row 1, column 1. Next, get 0s in the rest of column 1.

12. 7.3 Extending the Row Echelon Method to Larger Systems

13. 7.3 Extending the Row Echelon Method to Larger Systems
The corresponding matrix to the system is
Applying back-substitution we get the solution {(1, 2, –1)}.

14. 7.3 Reduced Row Echelon Method
Matrix methods for solving systems
Row Echelon Method form of a matrix
seen earlier
1s along the diagonal and 0s below
Reduced Row Echelon Method form of a matrix
1s along the diagonal and 0s both below and above

15. 7.3 Reduced Row Echelon Method
Example The augmented form of the system is
Using the row transformations, this augmented
matrix can be transformed to

16. 7.3 Reduced Row Echelon Method with the Graphing Calculator
Example Solve the system.
Solution The augmented matrix of the system is
shown below.

17. 7.3 Reduced Row Echelon Method with the Graphing Calculator
Using the rref command we obtain the row reduced
echelon form.

18. 7.3 Solving a System with No Solutions
Example Show that the following system is
inconsistent.
Solution The augmented matrix of the system is

19. 7.3 Solving a System with No Solutions
The final row indicates that the system is
inconsistent and has solution set .

20. 7.3 Solving a System with Dependent Equations
Example Show that the system has dependent
equations. Express the general solution using an
arbitrary variable.
Solution The augmented matrix is

21. 7.3 Solving a System with Dependent Equations
The final row of 0s indicates that the system has
dependent equations. The first two rows represent
the system

22. 7.3 Solving a System with Dependent Equations
Start with the augmented matrix

23. 7.3 Solving a System with Dependent Equations
The equations that correspond to the final matrix are

24. 7.3 Solving a System with Dependent Equations
Solving for y we get
Substitute this result into the expression to find x.
Solution set written with z arbitrary: