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

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

2. Concepts and Objectives
⚫ Determinant Solutions 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 4x x
− − =2
3 4 0x x
( )( )− + =4 1 0x x
= −4, 1x

9. Determinants
⚫ To calculate the determinant of a 33 matrix, repeat the
first two columns to help you draw the diagonals:
⚫ Again, your calculator can also calculate the determinant
of a matrix you have entered.
− −
−
8 2 4
7 0 3
5 1 2
−
−
−
−
−
= 7 0
5
8 2
1
4
3
8 2
7
2 5
0
1
=500= ( )30+ − 28+ (0− ( )24+ − ( ))28+ −

10. 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.

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

12. 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 = = − = −
71
1
8 7 15
8
xD = = − − = −
−
( )
15
6
5 6 11
1
yD = = − − =
−
−
= = =
−
15
7.5
2
xD
x
D
= = = −
−
11
5.5
2
yD
y
D

13. Classwork
⚫ College Algebra & Trigonometry
⚫ Page 874: 6, 8, 16-26 (even); page 849: 32-40 (even);
page 789: 38-56 (even)