In 1980 the U.S. population, broken down by regions, was...

In 1980 the U.S. population, broken down by regions, was $49.1$ million in the Northeast, $58.9$ million in the Midwest, $75.4$ million in the South, and $43.2$ million in the West. ${ }^{3}$ In 1990 the population 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. Set up the population figures for each year as a row vector, and then show how to use matrix operations to find the net increase or decrease of population in each region from 1980 to 1990 .

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Key Concepts

Matrix Operations

Matrix operations refer to the arithmetic processes that can be performed on matrices, including addition, subtraction, and multiplication. These operations allow for the systematic calculation and transformation of data sets. In the context of comparing two time periods, matrix operations provide a structured technique for handling many data points in parallel, enabling the computation of differences across corresponding components.

Element-wise Subtraction

Element-wise subtraction is an operation where two matrices or vectors of the same dimension have their corresponding elements subtracted from each other. This operation is particularly useful for determining the change between two data sets—in this case, assessing the net increase or decrease in population for each region from one time period to another. It allows for a straightforward comparison of individual components within structured data.

Row Vectors

A row vector is a 1 × n matrix, where n represents the number of elements. It is used to represent a set of related data, such as measurements or statistics from different categories or regions, arranged in a single row. This representation provides a compact and convenient way to handle and manipulate multiple data points simultaneously in matrix operations.

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