Calculate determinants of 2x2 and 3x3 matrices Use Cramer's Rule to solve linear systems.

1. 9.4 Cramer’s Rule
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. 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
= −

4. Determinants
⚫ Example: Find the determinant of
−
 
 
 
2 2
3 1

5. Determinants
⚫ Example: Find the determinant of
−
 
 
 
2 2
3 1
( )( ) ( )( )
−
= − −
2 2
2 1 3 2
3 1
= + =
2 6 8

6. Determinants
⚫ Example: Solve for x:
=
3
4
x
x x

7. 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

8. Determinants
⚫ To calculate the determinant of a 33 matrix, repeat the
first two columns to help you draw the diagonals:
⚫ Manually calculating the determinant of a matrix larger
than 3×3 is considerably more complicated, and is really
beyond the scope of this class.
− −
−
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
+ −

9. Determinants (cont.)
⚫ Desmos.com/matrix calculator can also calculate the
determinant of a matrix you have entered.

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

11. 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.

12. Cramer’s Rule
⚫ Example: Solve the system using Cramer’s Rule.
+ = −


+ =

5 7 1
6 8 1
x y
x y

13. 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