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College Algebra: An Introduction to Inverses


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:39
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 72 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Systems of Equations (33 lessons, $44.55)
College Algebra: Inverses and Matrices (5 lessons, $7.92)

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 now when you think about just regular numbers, okay, when you think about multiplication, the opposite of multiplication is division. And how does that work exactly? Well if you have a number like 5, it turns out that there's a special number that, if I multiply it by 5, I still get 5. And that special number is called the multiplicative identity, a.k.a. 1. So in fact, 1 has the property that, if you multiply it by any number, whether on the right or the left, you still get that number. And as a result, you can ask for the multiplicative inverse, the inverse of 5, which would be . And why is the inverse a ? Because if I take 5 and I multiply it by , what I see is the identity, 1. So these two numbers are multiplicative inverses, because their product gives me the multiplicative identity, 1. The identity has the property of 5 times the thing, which gives me 5. The thing times the 5 gives me 5.
Well now I want to start taking a look at the analogues of these ideas with square matrices. So the first idea is what does the identity matrix look like? That would be the matrix, which has the property that, if I multiply it by another matrix, I get back the other matrix. The identity doesn't change the value of the matrix. And then, the follow up question is how do you find the inverses? How do you have the analogue of in this context?
Okay, well let me first of all tell you what the identity matrix looks like. So the analogue of 1, so the identity matrix is a matrix that just has ones along the diagonal, and zeros everywhere else. Zeros everywhere, except along the main diagonal there, in which case you have ones. So for example, a two by two identity matrix would look like this, ones along the diagonal, zeros everywhere else. Is that really an identity matrix? Well let's see. Let's multiply it by a two by two matrix and see what happens. So let's take 3, 5, 5, 8, and do matrix multiplication, and see if we actually end up with this again. If this is supposed to act like 1, then 1 times anything should give me the anything again. Let's see what we get. Well remember how matrix multiplication works? I sort of do this kind of thing. So I take 3 x 1 + 0 x 5. Well that's 3. To find out what goes here, I take this row with this column, and I see 1 x 5 + 0 x 8. So I see 5. To get the second row, first column, I go to the second row, first column. So I do this activity, which is 0 + 5, which is 5. And finally I do 0 + 8, which is 8. And look, this is the same as that. So in fact, this does act like an identity when you multiply.
Okay, cool, so there's the identity matrix. It's the matrix that just has ones along this diagonal and 0 everywhere else. What about inverses? Well let me show you what an inverse would look like. In fact let me just start anew with an example. Let's multiply these people out. Let's take 3, 5, 5, 8, and let's multiply it by -8, 5, 5, -3. This is a completely different matrix you'll notice. But let's just do the matrix multiplication and see what we get. Here I see 3 x -8, which is -24. And then I add 5 x 5, which is 25. So I have -24 + 25. That's just 1. What would I have here? Well I do this thing. I see 3 x 5, and then a 5 x -3, so that's 15 + -15. That's 0. What do I have here? Well I do this and I see 5 x -8, which is -40 plus 8 x 5, which is 40. So -40 + 40 = 0. And here I have 25 - 24, which is 1. So look, I get the identity matrix. That means that this matrix must be the inverse matrix of this one. And why, because their product gives me the identity; just like with numbers, is the inverse of 5, because x 5 equals the identity, 1. In this case the identity matrix looks like this. And so in order to actually have this be the inverse, we must have the product of these two things actually give me this.
Well the question now is how do you actually find the inverse? It's not just the reciprocal of every single element there. So how do you actually find the inverse of a matrix? And what's sort of the analogue of like 0? You know you can't find the multiplicative inverse of 0, right, because 1 over 0 is undefined. Well it turns out the analogue of that here is whether the matrix is singular or not. Now remember a matrix is singular if its determinant is 0. And it turns out that the only matrices that have inverses are those that are nonsingular. So the only matrices that have inverses are those for which the determinant is not 0. So before, like you know you can't divide by 0, the same thing here. You can't have an inverse of a matrix whose determinant is 0. But notice the determinant of this, that's easy to see, it's 3 x 8, which is 24. And then I subtract off 25, so the determinant of this is -1, which is not 0, so it should have an inverse. And in fact, I just happen to know what it is. It equals this. So you can tell if a matrix is invertible or not, has an inverse, by just looking at its determinant. If its determinant is 0, it cannot be inverted. You cannot find its multiplicative inverse. And if the determinant is not 0, then you can find its inverse. The question now is how. How did I know this is the right matrix that's the inverse of this one? Coming up next, I'll show you the secret for the two by two case, and then later I'll show you the secret for the three by three case and even higher. I'll see you there.
Systems of Equations
Inverses and Matrices
An Introduction to Inverses Page [2 of 2]

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