Chapter 5: Matrices
and Determinants
Section 5.7: Determinants and Cramer’s Rule
Section 5.7: Determinants and
Cramer’s Rule
 Goal: To evaluate the determinant of a 2 x 2 or a 3 x 3
matrix and to solve systems of equations using Cramer’s
Rule
Section 5.7: Determinants and
Cramer’s Rule
 Determinant: a real number that can be associated with
any square matrix
The determinant of a 2 x 2 matrix
be
a1 b1 is defined to
a 2 b2
a1 b1 = a1b2 – a2b1
a 2 b2
The elements of a determinant are enclosed within vertical
bars, not brackets
Section 5.7: Determinants and
Cramer’s Rule
 Examples:
Find the determinant of the following:
1.
4 -2
3 -5
2.
0
0.6
-0.7
0.9
Section 5.7: Determinants and
Cramer’s Rule
 The determinant of a 3 x 3 matrix is defined to be:
a1 b1 c1
a2 b2 c2 = a1b2c3 + a2b3c1 + a3b1c2 – a1b3c2 – a2b1c3 – a3b2c1
a3 b3 c3
Section 5.7: Determinants and
Cramer’s Rule
 Shortcut: Use Expansion by Minors!
 Minor: the minor of an element in a 3 x 3 determinant is the
2 x 2 determinant that is found by eliminating the row and
column that contain that element
a1 b1 c1
a2 b2 c2 = a1 b2 c2 – b1 a2 c2 + c1 a2 b2
a3 b3 c3
b3 c3
a3 c3
a 3 b3
Section 5.7: Determinants and
Cramer’s Rule
 Examples: Evaluate each determinant using minors
3.
4 0 2
4.
21 6 9
-1 3 1
10 0 0
0 2 5
4
1 5
Section 5.7: Determinants and
Cramer’s Rule
 Homework:
 Practice Exercises Pg. 237 #1-9 (all)
Section 5.7: Determinants and
Cramer’s Rule
 Cramer’s Rule: a method of solving a system of equations
using determinants
 D is the determinant of variable coefficients
 Dx is the determinant when the x coefficients are replaced
by the constants
 Dy is the determinant when the y coefficients are replaced
with the constants
 Dz is the determinant when the z coefficients are replaced
with the constants
Section 5.7: Determinants and
Cramer’s Rule
 Then (x, y, z) is found by: ( D x , D y , D z )
D D D
Examples: Solve using Cramer’s Rule:
11x + 15y = -20
14x + 12y = 10
Section 5.7: Determinants and
Cramer’s Rule
 Examples: Solve using Cramer’s Rule
3x + y + 2z = 13
x+z=0
x – 2y + z = -8
Section 5.7: Determinants and
Cramer’s Rule
 Classwork: Practice Exercises Pg. 237 #11, 13, 21, 23
 Homework: Practice Exercises Pg. 237 #10, 12, 14, 22