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College Algebra: When can you Multiply Matrices?


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:11
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 56 MB
  • Posted: 06/26/2009

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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So the romantic question on the table is, can they multiply? I'm going to show you a whole bunch of matrices in pairs and you have to tell me whether, in fact, they can multiply or not. Let's start in right now and see.
For example, what about these two people, they look pretty much the same to me, can they multiply? No, because remember, the columns here have to equal the rows here. And so, in fact, this doesn't work out, I run out of things. If you try to do the method you fail right? You can't, right? Try it again. Do you see how the columns here have to equal the rows here? I have to be able to match them up in a one to one manner you see. What about this way? No, it still doesn't work. So this is not good, there's not a match so we have no match there. What about here? Does this match? Let's see. Well, do the number of columns here equal the number of rows here? No, try it. It doesn't work.
What about this way though? Will this way work? Let's try it. No, no good. So you've got to have the columns on the left equaling the number of rows on the right, so it's not a match. How about this one? Can these be put together and multiplied? Let's try it. This is great. And what's the answer going to be? Well, since this is a 3 here, this is going to be a 3 and this is a 3 because I take the number of rows here and the number of columns here and that tells me the size of the answer. This will be a 3 by 3. What about this one? Can they do it in this position? Well, let's try it. Oh, yes. Oh, this is great. And what would the answer be? Well, since the columns here matching up with the rows here, I'd take the row answer here by the column answer here so this is actually going to give me a 2 by 2, and can you see it? I actually want you to visualize this. It's a 2 by 2.
All right. Let's see what else we've got. Here's a big one. Can these two people do it? Well, let's see. No, it's not going to be good. What about this way though? Can it be done this way? Let's see. Oh, yes. So, in fact, this can be done because I have three columns here, which matches up with three rows here. What's the outcome going to be? It's going to be a 2 by 4, but this way won't work. So you see, because here I've got four columns and only two rows, they're not equal. How about this right here? Compatible? Let's see. No, it doesn't work and that's because this has four columns and this has only three rows, it's not going to work. What about this way though? Would this way work? Let's try this one. See, I run out of people right away, I only have one person there. It's not going to work, this person, no match. Let's see, what about this here? Do these two people go together? Let's see if they can multiply. I have no more places to go here and I have to keep going down here so that doesn't work.
What about this way? Let's try this. This works great because notice the number of columns here equals the number of rows here. What would the net gain be, the net outcome be? Well, it would be a 3 by 1, so in fact, this would be a 3 by 1. Let's actually figure out what it would equal. So the first spot is going to be 2 x 1 = 1 x 3 + 0 x -1, so that's 2 + 3 + 0 = 5. Then the second spot I jump down, 3 - 3 - 4 so it's a -4 when I add those numbers up. Remember what I do, I multiply each term and then add up those answers, so here I see a 1, a 3 and a -2, so it's going to be a 1 and a 4 and then a -2 is just a 2. So notice that my answer is a 3 by 1 just like I predicted because I have three things here and one thing here.
So remember that you can only multiply things when the number of columns on the left equals the number of rows on the right, so you can do this process perfectly. And then the outcome will be the number of rows you have here by the number of columns you have here.
All right. Try to practice multiplication and remember it's tricky because you have to have these things line up just right.
Systems of Equations
Multiplying Matrices: Can They Multiply? Page [1 of 1]

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