Math 111
Week 6 Homework
Due 2/16/2016
9.5 Problems #75 to 80
75. Write the matrix here:
76. Answer the question. _4______ Describe how
the matrix shows that this page can be reached
by one click from any other web page:
The four 1s in column 4 show that page 4 can
reach each of the other pages in a single click.
77. Write 𝐴2 here:
78. Answer the question. ____________________
Pages 1 and 2 cannot be reached using two
click path from any other page in the network.
79. __There are two different 2-click paths from page 2 to page 3._
80. __You can get from pge 4 back to page 4 in 2 clicks._
9.6 #72 on page 779
(a) Nothing to write here, just read the traffic map and see why the equations work.
(b) Show the matrix equation:
Show what you enter in your calculator to solve this system:
You can show either of these:
𝑥1
1
0 0 0
1
0 0 0
5
5
𝑥2
−1
1 0 0
−1
1 0 0
−3
−3
[
]∙[𝑥 ] = [ ]
[
] → [𝐴] [ ] → [𝐵] , [𝐴]−1 ∙ [𝐵] =
0 −1 1 0
0 −1 1 0
3
3
3
𝑥
0
0 1 1
4
0
0 1 1
11
11
or
1
0
0
1
1
0
[
1
1
1
−1 −1 −1
𝑥1
0
5
𝑥
0
−3
2
]∙[ ]=[𝑥 ]
0
3
3
𝑥4
1
11
Show the matrix result:
5
2
[ ]
5
6
(c) What does your result mean about traffic?
Between intersection A and B there are 5 cars per
minute, 2 cars per minute between B and C, 5 cars
per minute between C and D and 6 cars per
minute between A and D.
9.7 #36 on page 787
Create three triangles, and yours may be different than mine.
The left-hand triangle gives first matrix, the middle triangle
gives the middle matrix, and the last triangle goes with the
third matrix. After saving each matrix, I completed the
calculation below on my calculator:
Left Triangle:
Middle Triangle:
Right Triangle:
The area of the shaded pentagon is 26. No units were given.
9.7 Extended and Discovery #10
Show the matrix before you take the determinant. And I am going to show my work:
𝒙𝟐 + 𝒚𝟐 𝒙 𝒚 𝟏
𝟏 𝟎 𝟏] = 𝟎
𝐝𝐞𝐭 [ 𝟏
𝟓
−𝟏 𝟐 𝟏
𝟏𝟑
𝟑 𝟐 𝟏
𝒙𝟐 + 𝒚𝟐
𝐝𝐞𝐭 [ 𝟏
𝟓
𝟏𝟑
𝒙
𝟏
−𝟏
𝟑
𝒚
𝟎
𝟐
𝟐
𝟏
𝟏]
𝟏
𝟏
𝟏 𝟎 𝟏
𝟏 𝟎 𝟏
𝟏
𝟏 𝟏
𝟏
𝟏 𝟎
= (𝒙𝟐 + 𝒚𝟐 ) [−𝟏 𝟐 𝟏] − 𝒙 [ 𝟓 𝟐 𝟏] + 𝒚 [ 𝟓 −𝟏 𝟏] − 𝟏 [ 𝟓 −𝟏 𝟐]
𝟑 𝟐 𝟏
𝟏𝟑 𝟐 𝟏
𝟏𝟑 𝟑 𝟏
𝟏𝟑 𝟑 𝟐
=𝟎
= (𝒙𝟐 + 𝒚𝟐 )(−𝟖) − 𝒙(−𝟏𝟔) + 𝒚(𝟑𝟐) − 𝟏(𝟖) = 𝟎 Evaluating the determinants
−𝟖𝒙𝟐 + 𝟏𝟔𝒙 − 𝟖𝒚𝟐 + 𝟑𝟐𝒚 − 𝟖 = 𝟎
Reorganizing in order to complete the squares
𝒙𝟐 − 𝟐𝒙 + 𝒚𝟐 − 𝟒𝒚 + 𝟏 = 𝟎
Divide both sides by – 𝟖.
(𝒙𝟐 − 𝟐𝒙 + 𝟏) + (𝒚𝟐 − 𝟒𝒚 + 𝟒) = 𝟒
Complete both squares. Set equal to constant
What is the equation of the circle?
(𝒙 − 𝟏)𝟐 +(𝒚 − 𝟐)𝟐 = 𝟒
The circle has center (-1, -2) and the radius is 2.