# 10.1 Multiplying matrices

 Page 1 / 1
This module covers multiplication of matrices.

## Multiplying a row matrix by a column matrix

A “row matrix” means a matrix with only one row. A “column matrix” means a matrix with only one column. When a row matrix has the same number of elements as a column matrix, they can be multiplied. So the following is a perfectly legal matrix multiplication problem:

$\left[\begin{array}{cccc}1& 2& 3& 4\end{array}\right]x\left[\begin{array}{c}10\\ 20\\ 30\\ 40\end{array}\right]$

These two matrices could not be added, of course, since their dimensions are different, but they can be multiplied. Here’s how you do it. You multiply the first (left-most) item in the row, by the first (top) item in the column. Then you do the same for the second items, and the third items, and so on. Finally, you add all these products to produce the final number.

A couple of my students (Nakisa Asefnia and Laura Parks) came up with an ingenious trick for visualizing this process. Think of the row as a dump truck, backing up to the column dumpster. When the row dumps its load, the numbers line up with the corresponding numbers in the column, like so:

So, without the trucks and dumpsters, we express the result—a row matrix, times a column matrix—like this:

$\left[\begin{array}{cccc}1& 2& 3& 4\end{array}\right]\left[\begin{array}{c}10\\ 20\\ 30\\ 40\end{array}\right]=\left[\begin{array}{c}300\end{array}\right]$

There are several subtleties to note about this operation.

• The picture is a bit deceptive, because it might appear that you are multiplying two columns. In fact, you cannot multiply a column matrix by a column matrix . We are multiplying a row matrix by a column matrix. The picture of the row matrix “dumping down” only demonstrates which numbers to multiply.
• The answer to this problem is not a number: it is a 1-by-1 matrix.
• The multiplication can only be performed if the number of elements in each matrix is the same. (In this example, each matrix has 4 elements.)
• Order matters! We are multiplying a row matrix times a column matrix , not the other way around.

It’s important to practice a few of these, and get the hang of it, before you move on.

## Multiplying matrices in general

The general algorithm for multiplying matrices is built on the row-times-column operation discussed above. Consider the following example:

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\\ \text{10}& \text{11}& \text{12}\end{array}\right]$ $\left[\begin{array}{cc}\text{10}& \text{40}\\ \text{20}& \text{50}\\ \text{30}& \text{60}\end{array}\right]$

The key to such a problem is to think of the first matrix as a list of rows (in this case, 4 rows), and the second matrix as a list of columns (in this case, 2 columns). You are going to multiply each row in the first matrix, by each column in the second matrix. In each case, you will use the “dump truck” method illustrated above.

Start at the beginning: first row, times first column.

Now, move down to the next row. As you do so, move down in the answer matrix as well.

Now, move down the rows in the first matrix, multiplying each one by that same column on the right. List the numbers below each other.

The first column of the second matrix has become the first column of the answer. We now move on to the second column and repeat the entire process, starting with the first row.

And so on, working our way once again through all the rows in the first matrix.

We’re done. We can summarize the results of this entire operation as follows:

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\\ \text{10}& \text{11}& \text{12}\end{array}\right]$ $\left[\begin{array}{cc}\text{10}& \text{40}\\ \text{20}& \text{50}\\ \text{30}& \text{60}\end{array}\right]$ $=$ $\left[\begin{array}{cc}\text{140}& \text{320}\\ \text{320}& \text{770}\\ \text{500}& \text{1220}\\ \text{680}& \text{1670}\end{array}\right]$

It’s a strange and ugly process—but everything we’re going to do in the rest of this unit builds on this, so it’s vital to be comfortable with this process. The only way to become comfortable with this process is to do it. A lot. Multiply a lot of matrices until you are confident in the steps.

Note that we could add more rows to the first matrix, and that would add more rows to the answer. We could add more columns to the second matrix, and that would add more columns to the answer. However—if we added a column to the first matrix, or added a row to the second matrix, we would have an illegal multiplication. As an example, consider what happens if we try to do this multiplication in reverse:

$\left[\begin{array}{cc}\text{10}& \text{40}\\ \text{20}& \text{50}\\ \text{30}& \text{60}\end{array}\right]$ $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\\ \text{10}& \text{11}& \text{12}\end{array}\right]$ Illegal multiplication

If we attempt to multiply these two matrices, we start (as always) with the first row of the first matrix, times the first column of the second matrix: $\left[\begin{array}{cc}10& 40\end{array}\right]$ $\left[\begin{array}{c}1\\ 4\\ 7\\ \text{10}\end{array}\right]$ . But this is an illegal multiplication; the items don’t line up, since there are two elements in the row and four in the column. So you cannot multiply these two matrices.

This example illustrates two vital properties of matrix multiplication.

• The number of columns in the first matrix, and the number of rows in the second matrix, must be equal. Otherwise, you cannot perform the multiplication.
• Matrix multiplication is not commutative —which is a fancy way of saying, order matters. If you reverse the order of a matrix multiplication, you may get a different answer, or you may (as in this case) get no answer at all.

#### Questions & Answers

what is backward bend
Florence Reply
It's a curve that reverses direction
Ajibade
Good evening friends
Kojo
Evening
Ajibade
evening
Emmanuel
hi
Lawrence
Any Nigerian lady to meet here Well on eco ground
Suleiman
Good morning ooo I hope everyone is fine
Kojo
hello
James
yeah
Kojo
how?
Kojo
how time
James
cool
Kojo
Ok
James
it can can be described as when the demand is insufficient
Amaan Reply
high Dublin divino through Higginbotham
Shubham Reply
How does exchange rate affect the demand and supply?
austine
You mean demand and supply of currency ?
Jahangir
Hello
Hann
Hi
Jahangir
Am debby
Hann
am jahangir
Jahangir
how are you
Ogunsola
I am All fine Wbu ?
Jahangir
doing great
Ogunsola
Hw r u all
Mudassar
Am gud
Hann
Gd
Mudassar
Ogunsola
are you help for your country of economy leavel
Mutha
hi
Hammed
Hello
Assan
hi
Pranshi
Hello
Kamal
hi
Pranshi
hi
Abbas
hi
Abdirazak
ki sah botine
Neyshawn
hi
Nasir
Hello Economist
Nasir
hello
Victor
Hie... may anyone give me examples of countries that successfully implemented import substitution industrialisation.
Thabo
Nigeria
Junior
hey I'm emma from Uganda
niwagaba
Hello
Abraham
What is the difference between Demand and Quantity Demanded?
Weedh
Demand refers to how much a product is desired by buyer mean while quantity demanded refers to the amount of product a buyer is willing to buy at a certain price.
Junior
Good evening
Ajibade
Evening
Junior
evening
Ogunsola
someone wat is d textbook to read for banking and finance course
ayomide
Evening
Anjorin
Hw r u doing
Faku
yh
Benj
I'm good, u?
Anjorin
demand is the various quantity of goods or services a consumer is Willing and able to buy at a particular price over s given period of time while quantity demanded is the amount of goods or services being paid for
Florence
Good
Awudu
1.it allows producers to learn and invent 2.it allows agent to take advantage of economic scale 3.it allow agents to focus in areas of
oyebola Reply
can someone explain the motive of money to me?
Alemogho Reply
ask the question you don't know
ojo
opportunity cost
Sylvo
the motive of money is to pay for goods or services. it's also for transaction purposes bet it ordinary or unforseen
Silke
Money helps to reduce our scarcity problems. It can satisfy your wants to an extent
Okwori
How dose economics use science methods to explains it problems
Edmond Reply
according to Jhon Robinson the main problem of economics is what to produce? how to produce? whom to produce? economics tries to solve these problem scientifically by using its theories which are scientific in nature,basically micro and macro theories which just explain what the problem is...
Lavkesh
and now the solving techinique is also changed. only theories or its matematical +statistical version is not sufficient so the economist had developed Econometrics which includes economic theory+mathematical economics+statistics, for solving economic problem. This method treats economics as a pure
Lavkesh
..pure science so this technique is based on pure scientific methods.
Lavkesh
Economics is the scientific study of human behavior so there for science is used to solve economics problems using economics theories
Florence
economic is not higly subject of taminadu ...
Mutha
measurement of elasticity of demand
John Reply
Price elasticity of demand= Percentage change in quantity demanded/ Percentage change in price
Jale
For example, suppose that a 10 percent increase in the price of an ice-cream cone causes the amount of ice cream you buy to fall by 20 percent. We calculate your elasticity of demand as
Jale
Price elasticity of demand =20 percent /10 percent = 2
Jale
In this example, the elasticity is 2, reflecting that the change in the quantity demanded is proportionately twice as large as the change in the price.
Jale
what is economic
Samuel Reply
economics is a social science that studies the relationship of human behavior between ends and scare...
Lawrence
Why is economics as a subject is so important to human being?
austine
it's significant in study of managing available resources to satisfy human needs and wants
mohamed
economics is the science of wealth
Aamir
it is also said that it is the science of scarcity
Aamir
why study economic
Neo
Economic helps to meet our needs and wants nd to make choices.
Abraham
careers or jobs associated with economics
Neo
Economics is more than choices, scarcity, resources etc. It has a lot to do with psychology, social psychology. Understand motivation for consumer decisons.
TOM
okey
Neo
I what to study economic understand education what subject will I take in my jamb
DIRIOHA
English.mathematics.economics and one other social science subject
Ogunsola
economic is life start to end
Mutha
Plz help in lectures job for economics
Mudassar
acc to robbin . "Economics is a science that studies human behaviour as a relationship between scarce means and ends which have alternative uses"
Jahangir
Economics is important to man "human wants are insatiable
Hann
How does commercial banks create credit
Penda Reply
what is equilibrium price?
Stella Reply
happen when demand curve and supply curve meet
Enockz
intersection point of demand and supply curve
Sarjeel
the equilibrium price is that price at which the quantity demanded is equal to the quantity supplied... i.e the price at which what people want to buy is equal to what the sellers want to sell.
Che
what is difference between demand and quantity demand
Sarjeel
the price which is prevailing in the market ,it is the price at which the consumer is getting maximum satisfaction and the producer is getting maximum profit.
Rather
at equilibrium price demand for commodity is equal to supply of commodity..
Rather
dear sareel, quantity demand means the quantity of commodity which is demanded at a particular price e,g , your demand is for 4 oranges when price of orange is 5,, on the other hand demand means the whole quantity price relationship, quantity demand is shown on a particular point on demand curve,
Rather
sarjeel sbb, reply please...
Rather
what is elasticity of demand
Evans
what is Economics?
Ahmed
Economics can be defined as a social sciences that study human behavior as a relationship between end and scarce mean which have alternative use
Azeez
What is taxation
Azeez
economics is the study of scarcity
Suleiman
dear Suleiman how?
Enockz
economics is a subject matter that studies different economic activities as directed towards maximization of income at level of individual and maximization of social welfare at the level of society as a whole
Rather
economics studies us how to allocate scarce resources to full fill the maximum human wants as the resources have alternative uses..and r very scare to fullfill the human wants as the wants of humans are unlimited
Rather
demand is what a buyer is able and willing to buy at a specific price where by quantity demanded is amount of demand the buyer is able and willing to buy at a specific price.
daudi
thanks che..
Stella
how does commercial banks create credit
Penda
write models of Aggregate Supply?
abdi Reply
models of aggregate supply?
abdi
Then what's the law of elasticity?
austine
wat happen if money circulation failed to be controlled in any county?
Enockz Reply
the amount of money in circulation increases and cause aggregate demand to rise. Thus, hyperinflation.
Che
what's heteroscandacity
Crowther
price mechanism refers to the manner which price of goods and services affect the demand and supply of goods and services
Abaide
heteroscandacity: Is the circumstance where there is unequal variances across the observations.
Enockz
what is price mechanisms?
Che Reply
price mechanism refers to the manner which price of goods and services affect the demand and supply of goods and services
Abaide
what is demand
kabeer Reply
desire and ability to pay for a commodity is called demand.
Rather
i desire to become Prime Minister of India , is it demand ?please answer---
Rather
demand refers to the quantity demanded of a commodity that consumers are willing and able to buy at a particular price over a given period of time
Abigail
or the willingness power of a commoditythat is backed up by the ability to pay
Abigail
u r 100% right..
Rather
is ability to purchase a perticular goods at any price at a given time.
Neema
tanz Rather
Abigail
ur desire to become a prime minister of lndia is nt demand because u have nt made any efforts or willingness, it is called mere wish or lets say mere desire
Abigail
Thank you so much dear Abigail Mante for giving answer to my question..
Rather
U are welcome
Abigail
it's ok,dear..
Rather
The term ceteris paribus implies that
Oyelekan
Demand is the quantity demanded of a commodity at which a consumer is willing and able to purchase at various prices over a given period of time.
Suleman
ceteris parbus explains the law of demand,which states that at a higher price quantity demanded of a commodity is low and at a lower price quantity demanded of a commodity is high all other things being equal.
Abigail
U know right girl
Suleman
kk
Abigail
ceteris paribus is an economics terms used in situations where we considered other factors to remain unchanged
Ndeh
Thanks for the answer
Oyelekan
please how does disposable income affect demand?
Che
hell. somebody must tell me what is the meaning if cetrasparibus
Darling
plz. guys
Darling
the quantity of a commodity that a consumer ready willing and able to purchase at a particular price and in a particular period of time is known as demand
The
ceterusparibus means constant i,e,(other things do not change or other things remains constant)
Rather
wht
Bilal
right answer yawer saab
Sarjeel
things like what@Rather
Oyelekan
bilal sheikh why u has been written what
Sarjeel
e,g demand is determined by various factors like price of commodity, uses of good, expected changes in near future, nature of commodity , but when we learn price elasticity of demand , we keep other determinants of demand constant(i,e , we do not expect any change in price of good in near future,etc
Rather
i think u understand the term ceterus paribus..
Rather
bcz I can not understand
Bilal
what
Sarjeel
ok bro
Bilal
wht is other factors
Bilal
income of consumer, taste,
Sarjeel
Gold. Medlist in economics and. Lecturer of economics
HAZRAT
hello..what's are the factors affecting cross elasticity of demand.? at least five factors
Elzevery
The time it takes to get a product to market. That time limit has a name call "Just in Time."
William
thanks guys
kabeer
hey guys what is inelastic demand
Abaide
when there is no change in quantity demanded due to change in price then elasticity of demand is inelastic
Rather
Hi sucker how are you...
HAZRAT
thanks for the answer
Abaide
Any question about economics
HAZRAT
yes. what is the main cause of Heteroscedasticity
Abaide
hi
Habib
waht is elasticity of demand
Sarparah
elasticity of demand means change in quantity demanded divided by change in any of its determinent ,e,g , price ed= % change in quantity demanded/ % change in price.
Rather
elasticity of demand is the degree of responsiveness of quantity demanded of a commodity to a change in price of the commodity in question ,price of other related commodity n the consumers income
Abigail
yes
Kumako
ok
Abigail
guys what is monopoly
kabeer
A single seller in the market.
Tahir
what are the importance of economics
Eric Reply
what causes the demand curve to shift.
Stephen
other factors except price.it include test of a consumer fashen presence of substitute advatisment
Neema
factors other than price
Rather
Its helps to determine the price
Mavis
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Advanced algebra ii: conceptual explanations' conversation and receive update notifications?

 By By By