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Chapter 3 · Matrices

Example 1: Factory workers as a 3×23\times2 matrix

ExamplesWhat a matrix is · Sections 3.2 and 3.2.1
The question

Consider the following information regarding the number of men and women workers in three factories I, II and III

Men workersWomen workers
I3025
II2531
III2726
Exam answer

Taking the rows as factories I, II, III and the columns as men and women workers,

A=[302525312726]3×2A = \begin{bmatrix} 30 & 25 \\ 25 & 31 \\ 27 & 26 \end{bmatrix}_{3\times2}

a32=26a_{32} = 26 is the number of women workers in factory III.

Want the reason behind every step, the basics and the checks?

Represent the above information in the form of a 3×23 \times 2 matrix. What does the entry in the third row and second column represent?

The one idea this problem is built on

A matrix is a table with the labels taken off. The position of a number carries its meaning: the row tells you which factory, the column tells you which kind of worker. Once we agree on that layout, every entry aija_{ij} has exactly one meaning, and we can read it back by remembering what row ii and column jj stand for.

Given, to find, and the plan
Given
A table of 6 numbers: for each of 3 factories, the number of men workers and the number of women workers.
To find
(1) A 3×23\times2 matrix holding this information. (2) The meaning of the entry in row 3, column 2.
Plan
Decide what the 3 rows and the 2 columns stand for (the order 3×23\times2 forces the choice). Copy the numbers in, keeping each in its place. Then read off a32a_{32} using the row and column labels.

Basics you need

1What a matrix is

A matrix is an ordered rectangular array of numbers (or functions), written inside square brackets. The numbers are its elements or entries. Horizontal lines of entries are rows; vertical lines are columns. Basics: rows, columns and the address a_ij

2Order m×nm\times n

A matrix with mm rows and nn columns has order m×nm\times n (read "mm by nn"). Rows are always named first. So a 3×23\times2 matrix has 3 rows and 2 columns, and 3×2=63\times2=6 entries.

3The address aija_{ij}

aija_{ij} is the entry in row ii and column jj⁠: first index = row, second index = column. So a32a_{32} sits in row 3, column 2.

Solution

  1. Decide what the rows and columns stand for

    The table has 3 factories and 2 kinds of worker. The question asks for a 3×23\times2 matrix, that is 3 rows and 2 columns. So:

    • the 3 rows must be the 3 factories: row 1 = factory I, row 2 = factory II, row 3 = factory III;
    • the 2 columns must be the 2 kinds of worker: column 1 = men, column 2 = women.
    The order 3×23\times2 is a requirement on the shape: 3 rows, 2 columns. There are 3 factories and 2 kinds of worker, so only one assignment fits. Putting the kinds of worker in rows would give 2 rows and 3 columns, a 2×32\times3 matrix, which is not what was asked.
    In an order m×nm\times n⁠, the first number always counts rows and the second counts columns. Basics: rows, columns and the address a_ij

  2. Copy each number into its place

    With that layout, the row for factory I holds its men and women: 30 and 25. Factory II gives 25 and 31. Factory III gives 27 and 26. Entry by entry:

    a11=30,a12=25,a21=25,a22=31,a31=27,a32=26.\begin{aligned} a_{11} &= 30, & a_{12} &= 25,\\ a_{21} &= 25, & a_{22} &= 31,\\ a_{31} &= 27, & a_{32} &= 26. \end{aligned}

    Assembled:

    A=[302525312726].A = \begin{bmatrix} 30 & 25 \\ 25 & 31 \\ 27 & 26 \end{bmatrix}.

    It has 3 rows and 2 columns, so it is indeed of order 3×23\times2⁠, with 3×2=63\times2 = 6 entries, one for each number in the table.

    The matrix is just the table with the headings removed. Because the headings are gone, the only thing that tells us "25 in row 1, column 2 means 25 women in factory I" is the agreement we made in Step 1. That is why we fix the layout before writing any numbers.
    A matrix is an ordered array (Definition 1, Section 3.2): each entry's meaning comes from its position, so moving a number to another place gives (in general) a different matrix. Lesson: what a matrix is

  3. Read the entry in row 3, column 2

    The entry in the third row and second column is a32=26a_{32} = 26⁠. Row 3 stands for factory III and column 2 stands for women workers. So

    a32=26a_{32} = 26 is the number of women workers in factory III.

    MenWomen columns: type of worker Factory I Factory II Factory III rows: factory 30 25 25 31 27 26 row 3, column 2: 26 women workers in factory III 3 rows × 2 columns, so A has order 3 × 2
    The gold band is row 3 (factory III), the blue band is column 2 (women). The entry where they cross is a32=26a_{32} = 26⁠.
    To interpret an entry, translate its two indices back into words: the first index picks the row (here, factory III), the second picks the column (here, women). The number itself is a count of people, so the answer is "26 women workers in factory III", not just "26".
    First index = row, second index = column. Mixing them up would give a23a_{23} ⁠, which does not even exist in a matrix with only 2 columns. Basics: the address a_ij
Result A=[302525312726]3×2A = \begin{bmatrix} 30 & 25 \\ 25 & 31 \\ 27 & 26 \end{bmatrix}_{3\times2}

with rows = factories I, II, III and columns = men, women. The entry in the third row and second column, a32=26a_{32} = 26⁠, is the number of women workers in factory III. This agrees with the textbook's solution.

Check it with numbers

Every number of the table appears exactly once: the six entries are 30, 25, 25, 31, 27, 26, the same six numbers. Column 1 adds to 30+25+27=8230+25+27 = 82 men and column 2 adds to 25+31+26=8225+31+26 = 82 women, the same totals as the table's two columns, so no number has been moved to the wrong column. Row 3 adds to 27+26=5327+26=53⁠, the workforce of factory III.

Common mistakes
  1. Swapping rows and columns. Putting men and women in rows gives a 2×32\times3 matrix, not the 3×23\times2 one asked for.
  2. Reading a32a_{32} as row 2, column 3. The first index is the row. Row 2, column 3 does not exist here.
  3. Answering with a bare number. The question asks what the entry represents: say "the number of women workers in factory III".
What to take away
  1. A matrix stores information by position. Fix what each row and each column means before writing numbers.
  2. The order m×nm\times n is (rows) × (columns), and it decides which labels go along the side and which along the top.
  3. Interpreting aija_{ij} means translating ii and jj back into the row and column labels.

Every fact used, and where it comes from

FactUsed inSource
A matrix is an ordered rectangular array; rows and columnsSteps 1, 2Definition 1, Section 3.2 (Lesson: what a matrix is)
Order m×nm\times n = mm rows, nn columns; it has mnmn entriesSteps 1, 2Section 3.2.1
aija_{ij} lies in row ii⁠, column jjSteps 2, 3Section 3.2.1
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