1. Solving systems using matrices
4.4
Solving systems using matrices

2. “A” Matrix “The Matrix” A matrix is a rectangular array of numbers.
“The Matrix” is a movie with Keanu Reeves.
“The Matrix”

3. Example of a matrix Columns Rows Element
Note: A Square matrix has the same # of rows and columns

4. Writing an Augmented Matrix
Linear Equations 1:
Linear Equations 2:
Augmented Matrix
Note these are Standard Form

5. Writing an Augmented Matrix
Linear Equations 1:
Linear Equations 2:
Augmented Matrix
Note these are Standard Form
EX. 1

6. Writing an Augmented Matrix
Linear Equations 1:
Linear Equations 2:
Augmented Matrix
Write in Standard Form!!!
EX. 2

7. Row Transformations
All numbers in a row may be multiplied or divided by any nonzero real number.
You can replace rows by adding them to other rows and placing the sum in the row.

8. Transformations Example 1
All numbers in a row may be multiplied or divided by any nonzero real number.
Multiply R1 by -2 =

9. Transformations Example 2
All numbers in a row may be multiplied or divided by any nonzero real number.
Divide R2 by 3 =

10. Example 2 ANSWER

11. Transformations Example 3
All numbers in a row may be multiplied or divided by any nonzero real number.
Multiply R1 by 2 and multiply R2 by -4 =

12. Example 3 ANSWER

13. Transformations Example 4
You can replace rows by adding them to other rows and placing the sum in the row.
Replace R1 with R1+R2 =

14. Transformations Example 5
You can replace rows by adding them to other rows and placing the sum in the row.
Replace R2 with R1-R2 =

15. Example 5 ANSWER

16. Transformations Example 6
Replace R1 with : -2R1 + R2 =

17. Example 6 ANSWER
Note: R2 does not change!!!!

18. Transformations Example 7
Replace R2 with : -1/2R2 – R1 =

19. Example 7 ANSWER
Note: R1 does not change!!!!

20. Triangular form The 1’s and the 0 in these locations
a, p, and q are just constants

21. Use row transformation to get a matrix in triangular form
1.Work in column 1 to get the one.
2. Get the zero in column 1.
3. Get the 1 in column 2.
1st
2nd
3rd

22. Triangular form Example 1
Write the matrix in Triangular form =

23. Example 1 Steps 1st : 1/6 R1 2nd : Replace R2 with 10R1 + R2
3rd : -1/28 R2
Let’s Look at it !

24. Example 1 ANSWER

25. Triangular form Example 2
Write the Linear Equations in standard form.
Write the Augmented Matrix.
Get the matrix in Triangular Form.
Write the matrix back into Standard form.
Solve for x and y.

26. Put in Standard form.
2. Write the Augmented Matrix

27. 3. Try for Triangular Form.
4. Back to Standard Form.

28. 5. Solve for x and y. Looking here. Therefore ( 7/2 , -1)
y = -1, now substitute into equation 1.
x = 7/2
Therefore ( 7/2 , -1)
is where the lines cross.

29. Make sure to review these notes!