1. 3.6 SOLVING SYSTEMS
USING MATRICES

2. MATRICES
 A matrix is a rectangular array of numbers,
displayed within brackets.
2 4 1 
A=
 6 5 3

 The dimensions of a matrix are the numbers of rows
by the numbers of columns in the array.

3. MATRICES
 Each number in a matrix is a matrix element
and can be identified by its row and column
number
 Example:
 a11 a12 a13 
A=
 a21 a22 a23 


4. EXAMPLE: IDENTIFYING A
MATRIX ELEMENT
What is element a23 in matrix A?
 4 −9 17 1 
A= 0 5 8 6 
 
 −3 −2 10
 0 


5. SYSTEMS OF EQUATIONS AND
MATRICES
 We can represent systems of equations as matrices
 Each row represents an equation
 Each column represents the coefficients of a variable
 Example:

6. REPRESENTING SYSTEMS
WITH MATRICES

7. EXAMPLE: REPRESENT THE
SYSTEM WITH A MATRIX
x − 3y + z = 6

 x + 3 z = 12
 y = −5 x + 1


8. EXAMPLE: WRITE THE SYSTEM
OF EQUATIONS REPRESENTED
BY THE MATRIX
5 2 7 
 
0 1 9 

9. SOLVING A SYSTEM USING A
MATRIX
 We can solve a system by using a matrix and
performing row operations
 Row Operations are the “legal moves and
manipulations” we can make in a matrix
 Solving a system using row operations is similar
to elimination, because we use the same steps,
but don’t have variables

10. SOLVING A SYSTEM USING
MATRICES
 Row Operations:
 Switch any two rows
 Multiply a row by a constant
 Add (subtract) one row to another row
Make sure you write down what you are doing!

11. SOLVING A SYSTEM USING
MATRICES
 Goal:To use row operations to get a matrix
in the following forms:
1 0 0 a 
1 0 a   
  or 0 1 0 b 
0 1 b  0 0 1 c 
 
 Matricesthat represent the solution of a system
are in reduced row echelon form.

12. SOLVE THE SYSTEM OF
EQUATIONS USING A MATRIX
 x + 4 y = −1

2 x + 5 y = 4

13. SOLVE THE SYSTEM OF
EQUATIONS USING A MATRIX
9 x − 2 y = 5

3 x + 7 y = 17

14. SOLVE THE SYSTEM OF
EQUATIONS USING A MATRIX
 x + 2 y = 16

3 x + y = 8

15. ASSIGNMENT
 Page 179
 #8 – 11, 13 – 23 odd, 24, 27 – 29

16. 3.6 SOLVING SYSTEMS
USING MATRICES
Part 2 – Three- Variable Systems

17. USING MATRICES FOR THREE
VARIABLE SYSTEMS
 Same goal and row operations used to solve a
system with two variables

18. SOLVING A SYSTEM USING
MATRICES
 Row Operations:
 Switch any two rows
 Multiply a row by a constant
 Add (subtract) one row to another row
Make sure you write down what you are doing!

19. SOLVING A SYSTEM USING
MATRICES
 Goal:To use row operations to get a matrix
in the following forms:
1 0 0 a 
1 0 a   
  or 0 1 0 b 
0 1 b  0 0 1 c 
 
 Matricesthat represent the solution of a system
are in reduced row echelon form.

20. SOLVE THE SYSTEM OF
EQUATIONS USING A MATRIX

21. SOLVE THE SYSTEM OF
EQUATIONS USING A MATRIX

22. SOLVE THE SYSTEM OF
EQUATIONS USING A MATRIX

23. ASSIGNMENT
 3.6 Worksheet