Transcript Section 3.5 - Applications of Matrices and Determinants

Section 3.5 – Applications of Matrices and Determinants
Pick Up Worksheet From Your Folder
Application 1 – Areas of Plane Figures
1. Find the area of the triangle whose vertices are
(5, 2), (7, 1), (-2, 3)
5
2 1
5 2
5
1 7 1
1
7 1 1  1
1
1
2
2  2 3
2 3
7
2 3 1
1
 1 23   119   1 9   
2
5
The area of the triangle is
2
2

1
5
2
2. Find the area of the triangle whose vertices are:
(-2, 1), (0, 4), (3, 6)
2 1 1
1
0
2
3
5
1
4 1  1 12   1 15   1 8   
2
2
6 1
5
The area of the triangle is
2
3. Find the area of the parallelogram whose vertices are:
(2, 6), (1, -3), (-2, 4), (-3, -5)
2
6
1
1 3 1  34
2 4 1
2
4
1
1 3 1  34
3 5 1
The area of the parallelogram is 34.
4. Find the area of the parallelogram whose vertices are:
(7, 1), (0, 4), (6, 2), (-1, 5)
7 1 1
0 4 1  4
6 2 1
The area of the parallelogram is 4.
Application 2 – Collinearity of Points
5. Use a determinant to determine whether the points
(2, 7), (-3, -3), (5, 13) are collinear.
2
7
1
3 3 1   24   1 9   115   0
5 13 1
Since determinant is zero, the three points are collinear
6. Use a determinant to determine whether the points
(1, -2), (-4, 1), (0, 3) are collinear.
1
2 1
4
0
1
3
1   12  1 3   1 7  22
1
Since the determinant is non-zero, the three points are NOT
collinear. They form a triangle whose area is 3.
Application #3 - Cryptology
 2 1
1
,
use
A
to decode:
7. If A  

4 3
27 75 23 65 77 177 32
78
BA 1   109.5 61.5
Let B  27 75
27 
 3 C
1
Let B    A B   
75 
21 U
 77  1
27  _
Let B  
A B 

177 
23  W
23 
 2 B
1
Let B    A B   
65 
19  S
32 
9I
1
Let B    A B   
78 
14  N
CUBS WIN
1
 2

8. Use A   2

6
 3
0 2 1
1 1 0
1
4
3
1
2
5
2
0
0
4
3

5  to encode HOMECOMING.

2
1 
 8   1 
15   5 
  

A 13   113 
  

5
108
  

 3   9 
1  5 113 108 9 39
15   39 
13   5 
  

A  9   156 
  

14
177
  

 7   12
 5 156 177
 12
Application #4 - Networking
9. Refer to worksheet for the problem.
2
a.
1
3
4
b.
0
0
A
1

0
1 0 1
0 1 0

1 0 1

0 1 0
c.
Exactly one relay = A 2
0
1

0

1
d.
0 2 0
1 0 1

1 2 1

1 0 1
At most one relay = A 2  A
0
1

1

1
1 2 1
1 1 1

2 2 2

1 1 1
Application #5 – Transition Matrices
10. Refer to worksheet for the problem.
I
B
1500
500
I
B
I .8 .2 
B  .1 .9 
.8 .2
 1250 750
a. 1500 500 

.1 .9 
3
1250 Iphones
750 Blackberries
.8 .2 
 952.5 1047.5
b. 1500 500 

 .1 .9 
952 Iphones
1048 Blackberries