Problems 61-70 refer to the following transition matrix P and its powers A B C P = A B C .6 .3 .1 .2 .5 .3 .1 .2 .7 A B C P 2 = A B C .43 .35 .22 .25 .37 .38 .17 .27 .56 A B C P 3 = A B C .35 .348 .302 .262 .336 .402 .212 .298 .49 Using a graphing calculator to compute powers of P , find the smallest positive integer n such that the corresponding entries in P n and P n + 1 are equal when rounded to three decimal places.
Problems 61-70 refer to the following transition matrix P and its powers A B C P = A B C .6 .3 .1 .2 .5 .3 .1 .2 .7 A B C P 2 = A B C .43 .35 .22 .25 .37 .38 .17 .27 .56 A B C P 3 = A B C .35 .348 .302 .262 .336 .402 .212 .298 .49 Using a graphing calculator to compute powers of P , find the smallest positive integer n such that the corresponding entries in P n and P n + 1 are equal when rounded to three decimal places.
Solution Summary: The author explains how to approximate the smallest positive integer n such that the entries of
Problems 61-70 refer to the following transition matrix
P
and its powers
A
B
C
P
=
A
B
C
.6
.3
.1
.2
.5
.3
.1
.2
.7
A
B
C
P
2
=
A
B
C
.43
.35
.22
.25
.37
.38
.17
.27
.56
A
B
C
P
3
=
A
B
C
.35
.348
.302
.262
.336
.402
.212
.298
.49
Using a graphing calculator to compute powers of
P
, find the smallest positive integer
n
such that the corresponding entries in
P
n
and
P
n
+
1
are equal when rounded to three decimal places.
Q.1 Suppose a simple economy with only an agriculture industry and a
steel industry with the following technology matrix. Find the gross
production of each industry if surpluses of 15 units of agriculture products
and 35 units of steel are desired.
A S
0.3 0.2 Agrikultwe
0.1 0.4] Steel
A =
Next question
Dem. Rep.
In a certain town, the proportions of voters
voting Democratic and Republican by various
age groups is summarized by matrix A, and the
population of voters in the town by age group is
given by matrix B.
0.28 0.72
Under 30
0.78 0.22 | = A
30-50
Over 50
0.35 0.65
Interpret the entries of the matrix product BA.
| 1000
7000
6000
B =
Under 30
30-50
Over
50
In the matrix BA, the first entry means that there are
voters
and the second entry means that there are
voters
SECTION I: MATRIX OPERATIONS
[2
-1
1.5
A =
2.75
1
-1
-3
-0.5
4
0.25
0.1
0.2
-15.2
0.6
D =
0.05
[sym.
7
2
-10
0
0.25
0
1
1
E =
-15.575
1.975 2
1.4
-0.9
1.075 1.2 5.9
1. Find the TRANSPOSE of B.
Name it, "Matrix F".
2. Find the PRODUCT of Matrices A and F.
Name it, "Matrix G".
3. Matrix D is symmetric. Find the SUM of Matrices G and D.
Name it, "Matrix H".
4. Find the DIFFERENCE between Matrices H and E. That is: [H] - [E].
Name it, "Matrix I".
5. AUGMENT Matrix C with Matrix I. Write them in both in MATRIX AND EQUATION
FORM. Remember: Ax=B, or in this case, Ix=C.
Use the variables: w, x, y and z when writing them in equation form.
6. Using the formula discussed in class, determine if Matrix is DIAGONALLY
DOMINANT. If yes, proceed to section 2. If not, rearrange Matrix I so that it becomes
diagonally dominant. Since we have previously augmented matrix I with C, rewrite
the system of linear equations (just as with Item 5) with the CORRESPONDING
rows from matrix C both in MATRIX AND…
Chapter 9 Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Mathematics with Applications In the Management, Natural and Social Sciences (11th Edition)
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