Section 3.5 - Applications of Matrices and Determinants
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Transcript Section 3.5 - Applications of Matrices and Determinants
Section 3.5 – Applications of Matrices and Determinants
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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 23 119 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 9 115 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
9I
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