   Chapter 4.1, Problem 44E

Chapter
Section
Textbook Problem

# Population Movement In 1990 the U.S. population, broken down by regions, was 50.8 million in the Northeast, 59.7 million in the Midwest, 85.4 million in the South, and 52.8 million in the West.4 Between 1990 and 2000 the population in the Northeast grew by 2.8 million, the population in the Midwest grew by 4.7 million, the population in the South grew by 14.8 million, and the population in the West grew by 10.4 million. Set up the population figures for 1990 and the growth figures for the decade as row vectors. Assuming the same population growth from 2000 to 2010 as from 1990 to 2000, use matrix operations to estimate the population in each region in 2010. Compare the predicted 2010 population in the Northeast with the actual population given in Exercise 43.

To determine

To calculate: The population in each region in 2010. In addition, compare the predicted 2010 population in northeast with actual population given in exercise 43.

Explanation

Given Information:

The US population broken down by region was 50.8 million in the Northeast, 59.7 millions in the Northeast grew by 2.8 million, the population in the Midwest grew by 4.7 million, the population in in the South grew by 14.8 million, and population in the West grew by 10.4 million. Set up the population figure for 1990 and the growth figure for decade as row vector and assuming the same population growth from 2000 to 2010 as from 1990 to 2000.

Formula used:

When a matrix has m rows and n columns, then the matrix is said to have dimension of m×n.

For two matrices A and B with equal dimension, the sum of difference is simplified as,

(a11a12a21a22)±(b11b12b21b22)=(a11±b11a12±b12a21±b21a22±b22)

Calculation:

Consider the following data used to for calculation,

The US population broken down by region was 50.8 million in the Northeast, 59.7 millions in the Northeast grew by 2.8 million, the population in the Midwest grew by 4.7 million, the population in in the South grew by 14.8 million, and population in the West grew by 10.4 million. Set up the population figure for 1990 and the growth figure for decade as row vector and assuming the same population growth from 2000 to 2010 as from 1990 to 2000.

Write the populations in the year 1990 as row vectors.

Thus, population in 1990 is [50.859.785.452.8].

Write the populations in the year 2000 as row vectors.

Thus, population in 2000 is [53.664.4100.263.2].

Recall that a matrix with m rows and n columns is of dimension m×n, where m and n are positive integers.

Since, number of rows in [53.664.4100.263.2] is 1 and number of columns is 4,

Substitute 1 for m and 4 for n in m×n.

Thus, dimension of [53.664.4100.263.2] is 1×4.

Since, number of rows in [55.366.9114.671.9] is 1 and number of columns is 4,

Substitute 1 for m and 4 for n in m×n.

Thus, dimension of [55.366.9114.671.9] is 1×4.

Thus, both the matrices have same dimensions.

Consider, the net change in the population is the difference of population in 2000 and in 1990.

Thus, the net change is [53.664.4100.263.2][50.859.785.452.8].

Consider, [53.664.4100.263.2][50.859.785.452.8].

Recall that for two matrices A and B with equal dimension, AB is the difference of the corresponding entries of the matrices.

Simplify [53.664.4100.263.2][50.859.785.452.8].

[53

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