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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:

1 2 3 4 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} # 4{}} right ]} {} x [ 10 20 30 40 ]

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 picture illustrating how to multiply a matrix. In this example a 1x4 matrix is multiplied by a 4x1 matrix.

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:

A picture illustrating how to multiply a matrix. In this example a 1x4 matrix is multiplied by a 4x1 matrix.

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

[ 1 2 3 4 ] [ 10 20 30 40 ] = [ 300 ]

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:

1 2 3 4 5 6 7 8 9 10 11 12 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ##4 {} # 5 {} # 6 {} ## 7 {} # 8 {} # 9 {} ##"10" {} # "11" {} # "12"{} } right ]} {} 10 40 20 50 30 60 size 12{ left [ matrix { "10" {} # "40" {} ##"20" {} # "50" {} ## "30" {} # "60"{}} 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.

A picture showing the first step in multiplying matrices.

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

A picture showing the second step in multiplying matrices.

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.

A picture showing the following steps in multiplying matrices.

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.

A picture showing the following steps in multiplying matrices for the second row.

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

A picture showing the first step in multiplying matrices.

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

1 2 3 4 5 6 7 8 9 10 11 12 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ##4 {} # 5 {} # 6 {} ## 7 {} # 8 {} # 9 {} ##"10" {} # "11" {} # "12"{} } right ]} {} 10 40 20 50 30 60 size 12{ left [ matrix { "10" {} # "40" {} ##"20" {} # "50" {} ## "30" {} # "60"{}} right ]} {} = 140 320 320 770 500 1220 680 1670 size 12{ left [ matrix { "140" {} # "320" {} ##"320" {} # "770" {} ## "500" {} # "1220" {} ##"680" {} # "1670"{} } 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:

10 40 20 50 30 60 size 12{ left [ matrix { "10" {} # "40" {} ##"20" {} # "50" {} ## "30" {} # "60"{}} right ]} {} 1 2 3 4 5 6 7 8 9 10 11 12 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ##4 {} # 5 {} # 6 {} ## 7 {} # 8 {} # 9 {} ##"10" {} # "11" {} # "12"{} } 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: [ 10 40 ] 1 4 7 10 size 12{ left [ matrix { 1 {} ##4 {} ## 7 {} ##"10" } 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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Matrices. OpenStax CNX. Mar 31, 2011 Download for free at http://cnx.org/content/col11288/1.2
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