Calculate the determinant of a square matrix Use Cramer's Rule to solve a system of linear equations

1. 9.3 Determinant Solution of
Linear Systems
Chapter 9 Systems and Matrices

2. Concepts and Objectives
 Determinant Solution of Linear Systems
 Calculate the determinant of a square matrix
 Use Cramer’s Rule to solve a system of equations

3. Systems and Matrices
 A matrix is a rectangular array of numbers enclosed in
brackets. Each number is called an element of the
matrix.
 There are three different ways of using matrices to solve
a system:
 Use the multiplicative inverse.
 The Gauss-Jordan Method, which uses augmented
matrices.
 Cramer’s Rule, which uses determinants.

4. Determinants
 Every n  n matrix A is associated with a real number
called the determinant of A, written  A .
 The determinant is the sum of the diagonals in one
direction minus the sum of the diagonals in the other
direction.
 Example:
3 4
6 8
  
24 24 48
     
  
3 8 6 4
a b
c d
ad cb
 

5. Determinants
 Example: Find the determinant of

 
 
 
2 2
3 1

6. Determinants
 Example: Find the determinant of

 
 
 
2 2
3 1
     

  
2 2
2 1 3 2
3 1
  
2 6 8

7. Determinants
 Example: Solve for x:

3
4
x
x x

8. Determinants
 Example: Solve for x:

3
4
x
x x
 
2
3 4
x x
  
2
3 4 0
x x
  
  
4 1 0
x x
 
4, 1
x

9. Determinants
 To calculate the determinant of a 33 matrix, repeat the
first two columns to help you draw the diagonals:
 

8 2 4
7 0 3
5 1 2





 7 0
5
8 2
1
4
3
8 2
7
2 5
0
1
50
0
  
30
  28
 0
  
24
   
28
 

10. Determinants (cont.)
 Desmos.com/matrix has a calculator that can also
calculate the determinant of a matrix you have entered.

11. Determinants (cont.)
 Desmos.com/matrix has a calculator that can also
calculate the determinant of a matrix you have entered.

12. Cramer’s Rule
 To solve a system using Cramer’s Rule, set up a matrix of
the coefficients and calculate the determinant (D).
 Then, replace the first column of the matrix with the
constants and calculate that determinant (Dx).
 Continue, replacing the column of the variable with the
constants and calculating the determinant (Dy, etc.)
 The value of the variable is the ratio of the variable
determinant to the original determinant.

13. Cramer’s Rule
 Example: Solve the system using Cramer’s Rule.
  


 

5 7 1
6 8 1
x y
x y

14. Cramer’s Rule
 Example: Solve the system using Cramer’s Rule.
5
6 1
7 1
8
x y
x y
 


 


40 4
7
6 8
2 2
5
D     
7
1
1
8 7 15
8
x
D      

 
1
5
6
5 6 11
1
y
D     


  

15
7.5
2
x
D
x
D
   

11
5.5
2
y
D
y
D

15. Classwork
 9.3 Assignment (College Algebra)
 Page 874: 4-8, 16-26 (even); page 849: 32-40 (even);
page 480: 40-46 (even)
 9.3 Classwork Check
 Quiz 9.1