Section 8.5
Determinants and Cramer’s Rule
The Determinant of a 2 x 2
Matrix
Example
Evaluate the determinant of each of the following matices:
 2 3
a. 

5
1


 3 2 
b. 

4

1


Solving Systems of Linear
Equations in Two Variables
Using Determinants
Example
Use Cramer's Rule to solve the system:
2x-3y=-11
x+2y=12
Example
Use Cramer's Rule to solve the system:
3x+2y=-1
2x-4y=10
The Determinant of a
3 x 3 Matrix
Evaluate the determinant of the following matrix:
 2 1 0 
 1 1 2 


 3 1 0 
Example
Evaluate the determinant by hand, then check your
answer on the calculator.
 2 1 3
 3 0 1


 1 2 3
Solving Systems of Linear
Equations in Three Variables
Using Determinants
Example
Use Cramer's rule to solve:
-2x+y =1
x-y-2z=2
3x+y =6
Cramer’s Rule with Inconsistent
and Dependent Systems
The Determinant of Any
N x N Matrix
The determinant of a matrix with n rows and n columns is said
to be an nth-order determinant. The value of an nth-order determinant
can be found in terms of determinants of order n-1.
We can generalize the idea for fourth-order determinants and
higher. We have seen that the minor of the element a i j is the
determinant obtained by deleting the ith row and the jth column
in the given array of numbers. The cofactor of the element a i j
is (-1)i  j times the minor of the a ij th entry. If the sum of the row
and column (i+j) is even, the cofactor is the same as the minor.
If the sum of the row and column (i+j) is odd, the cofactor is the
opposite of the minor.
Example
Evaluate the determinant of the following matrix. Notice
that you can use either the third or the fourth columns.
1 2 0 0
 0 1 2 0 


 1 2 0 1



1
3
1
1


Evaluate the determinant
3 2
 1 4 


(a) 14
(b) 10
(c) 8
(d) 11
Use Cramer's Rule to solve the linear systems.
-x+2y=7
2x-2y=-4
(a) ( 1, 2)
(b) (2, 2)
(c) (3, 4)
(d) (3,5)