<< Chapter < Page
  Matrices   Page 1 / 1
Chapter >> Page >
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

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Matrices. OpenStax CNX. Mar 31, 2011 Download for free at http://cnx.org/content/col11288/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Matrices' conversation and receive update notifications?

Ask