Chapter 3 · Matrices

Example 19: Election campaign costs

ExamplesMatrix multiplication in a word problem · Section 3.4.5

The question

In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix A as

A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix}\ \begin{matrix} \text{Telephone} \\ \text{Housecall} \\ \text{Letter} \end{matrix}

The number of contacts of each type made in two cities X and Y is given by (columns Telephone, Housecall, Letter; rows X, Y)

B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10{,}000 \end{bmatrix}

Find the total amount spent by the group in the two cities X and Y.

Exam answer

$B$ is $2\times3$ and $A$ is $3\times1$ ⁠, so $BA$ is defined and is $2\times1$ ⁠. The amount spent (in paise) is $BA$ ⁠.

\begin{aligned} &\text{City X:} \\ &1000{\cdot}40 + 500{\cdot}100 + 5000{\cdot}50 \\ &= 40{,}000 + 50{,}000 + 250{,}000 \\ &= 340{,}000 \end{aligned}

\begin{aligned} &\text{City Y:} \\ &3000{\cdot}40 + 1000{\cdot}100 + 10{,}000{\cdot}50 \\ &= 120{,}000 + 100{,}000 + 500{,}000 \\ &= 720{,}000 \end{aligned}

\therefore\ BA = \begin{bmatrix} 340{,}000 \\ 720{,}000 \end{bmatrix}

Since 100 paise $=$ ₹1, these are ₹3400 and ₹7200.

Amount spent: ₹3400 in city X and ₹7200 in city Y.

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

The one idea this problem is built on

The cost in a city is (number of contacts of each type) $\times$ (cost of that type), added over the three types. That is exactly a row times a column: a row of $B$ (one city's contacts, by type) times the column $A$ (cost per contact, by type). The shared label, the type of contact, must be the index that gets summed, so it has to be the columns of the first factor and the rows of the second. That forces the product $BA$ ⁠.

Given, to find, and the plan

Given: $A$ ⁠: cost per contact in paise, order $3\times1$ (rows: telephone, house call, letter). $B$ ⁠: numbers of contacts, order $2\times3$ (rows: cities X, Y; columns: the same three types in the same order).
To find: The total amount spent in city X and in city Y.
Plan: Decide which product matches the meaning (and is defined), compute its two entries, then convert paise to rupees and state the answer in words.

Basics you need

1A table is a matrix with labels

Rows and columns of a matrix carry meanings. Here each row of $B$ is a city and each column a type of contact. Lesson: what a matrix is

2When a product is defined

$(m\times n)(n\times p)$ is defined and is $m\times p$ ⁠. Basics: rows, columns and the address

3Row times column

Pair up the entries in order, multiply, add. Basics: pair up, multiply, add

4Units

(contacts) $\times$ (paise per contact) = paise, and 100 paise = ₹1.

Solution

Read the two matrices with their labels

Telephone House call Letter

X: row 1 of $B$ 1000 500 5000

Y: row 2 of $B$ 3000 1000 10,000

Paise each: $A$ 40 100 50

$B$ is $2\times3$ (2 cities, 3 types) and $A$ is $3\times1$ (3 types, 1 cost each).

Lining up the labels first shows that the types appear in the same order, telephone, house call, letter, along the columns of $B$ and down the column $A$ ⁠. That shared order is what makes a row-times-column product meaningful.
Choose the product: $BA$ ⁠, not $AB$

$AB$ would be $(3\times\mathbf{1})(\mathbf{2}\times3)$ ⁠: inner sizes $1\ne2$ ⁠, so $AB$ is not defined.

$BA$ is $(2\times\mathbf{3})(\mathbf{3}\times1)$ ⁠: inner sizes $3=3$ ⁠, so $BA$ is defined and is $2\times1$ ⁠, one total for each city.

The arithmetic and the meaning agree. In $BA$ ⁠, the summed index is the columns of $B$ and the rows of $A$ ⁠, which is the type of contact: we add up over types, exactly as the cost of a city requires. The leftover labels are the rows of $B$ (cities) and the single column of $A$ (cost), so the answer is a column of costs, one per city.

Order of a product, Section 3.4.5. Lesson: matrix multiplication
Compute the two entries of $BA$

Row X of $B$ times $A$ ⁠, then row Y of $B$ times $A$ ⁠. Write $BA=[t_{i1}]$ ⁠:
$\begin{aligned} t_{11} &= 1000\cdot 40 + 500\cdot 100 + 5000\cdot 50\\ &= 40{,}000 + 50{,}000 + 250{,}000\\ &= 340{,}000\\ t_{21} &= 3000\cdot 40 + 1000\cdot 100\\ &+ 10{,}000\cdot 50\\ &= 120{,}000 + 100{,}000 + 500{,}000\\ &= 720{,}000 \end{aligned}$ $BA = \begin{bmatrix} 340{,}000 \\ 720{,}000 \end{bmatrix}\ \begin{matrix} \to X \\ \to Y \end{matrix}$

City X's row of contacts (gold) meets the column of costs (blue): matching types are multiplied and the three products added, giving 340,000 paise (green). City Y's row gives 720,000 paise the same way.

Each term is (contacts of one type) $\times$ (paise per contact of that type), so each term is in paise: for example $5000\cdot50=250{,}000$ paise is what the letters in city X cost. Adding the three terms gives the city's total.

$t_{i1}=b_{i1}a_{11}+b_{i2}a_{21}+b_{i3}a_{31}$ ⁠. Basics: pair up, multiply, add
Convert paise to rupees

Since 100 paise = ₹1, divide by 100:
$\begin{aligned} 340{,}000 \div 100 &= 3400\\ 720{,}000 \div 100 &= 7200 \end{aligned}$
So 340,000 paise is ₹3400 and 720,000 paise is ₹7200.

The costs were given in paise, so the product is in paise. Rupees are the natural unit for amounts this large, and the textbook states the answer both ways.
State the answer in words

The group spent 340,000 paise = ₹3400 in city X and 720,000 paise = ₹7200 in city Y. Together that is $3400+7200=10{,}600$ rupees.

"Total amount spent in the two cities X and Y" is answered city by city, as the textbook does; the combined figure is a useful extra.

	Telephone	House call	Letter
X: row 1 of $B$	1000	500	5000
Y: row 2 of $B$	3000	1000	10,000
Paise each: $A$	40	100	50

Result

BA = \begin{bmatrix} 340{,}000 \\ 720{,}000 \end{bmatrix}\ \text{paise}

So the group spent ₹3400 in city X and ₹7200 in city Y, as in the textbook.

Check it with numbers

Work by type instead of by city. Total telephone contacts: $1000+3000=4000$ ⁠, costing $4000\cdot40=160{,}000$ paise. House calls: $500+1000=1500$ ⁠, costing $150{,}000$ paise. Letters: $5000+10{,}000=15{,}000$ ⁠, costing $750{,}000$ paise. Altogether that is 1,060,000 paise, which equals $340{,}000+720{,}000$ ⁠, that is ₹10,600.

Common mistakes

Writing $AB$ ⁠. It is not defined here; the order of the factors follows the meaning, not the alphabet.
Forgetting the units. The product is in paise, because the costs are in paise. 340,000 is not ₹340,000.
Mixing up the type order. The columns of $B$ and the rows of $A$ must list the types in the same order, or the wrong costs get multiplied.

What to take away

In a word problem, the label shared by two tables (here, type of contact) is the one that gets summed; it must be the columns of the first factor and the rows of the second.
The labels left over (cities, and cost) become the rows and columns of the answer.
Always carry the units through: quantity $\times$ price per unit = money.

Every fact used, and where it comes from

Fact	Used in	Source
Matrices as labelled tables	Step 1	Section 3.2 (Lesson: what a matrix is)
Product defined iff inner sizes agree; order $m\times p$	Step 2	Section 3.4.5 (Lesson: matrix multiplication)
Row times column	Step 3	Section 3.4.5, definition of the product
100 paise = ₹1	Step 4	Indian currency

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Example 19: Election campaign costs

Basics you need

Solution

Read the two matrices with their labels

Choose the product: BABABA⁠, not ABABAB

Compute the two entries of BABABA

Convert paise to rupees

State the answer in words

Every fact used, and where it comes from

Choose the product: $BA$ ⁠, not $AB$

Compute the two entries of $BA$