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College Algebra: Inverses: 3x3 Matrices

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:40
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 93 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra Review (30 lessons, $59.40)
College Algebra: Systems of Equations (33 lessons, $44.55)
College Algebra: Inverses and Matrices (5 lessons, $7.92)

This lesson will show you how to take the approach you would use for calculating the inverse of a 2X2 square matrix in finding the inverse of a 3X3 square matrix. Once you find the inverse matrix, you should be able to multiply the original matrix by the inverse matrix and get the identity matrix. The identification of the inverse of 3X3 square matrices begins with finding the determinant (or scalar) of the full 3X3 matrix followed by finding the determinant of sub components of that 3X3 matrix (finding the minor determinants). Once all determinants are found, you'll apply a sign chart to the resulting 3X3 matrix and flip the matrix across its diagonal. The last step will be to multiply your result by the reciprocal of the determinant of the original 3X3 matrix. You would be able to use this same approach to find the inverse of a larger square matrix (4X4 or larger), but the calculation thereof would be very cumbersome.

For Professor Burger's lesson on finding inverses of 2X2 matrices, check out http://www.mindbites.com/lesson/806.

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

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 http://www.thinkwell.com/student/product/collegealgebra. 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

Thinkwell
Thinkwell
2174 lessons
Joined:
11/13/2008

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...

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Recent Reviews

Nopic_grn
what
01/03/2010
~ Janet2

this video length is about 8 minutes, and it stops about halfway through so I don't get to see all of it, which doesn't help me at all.

Nopic_tan
Thank You!
10/17/2009
~ Assessment

I am a phd student. This little short video did more for me in 30 minutes or less than any review I have attempted thus far in trying to remember how to find the Inverse of a Matrix! Excellent! Good job!

Nachan_homepage
Matrices are easy!
01/16/2009
~ nachan

Professor Burger teaches all the aspect of matrices. This particular lesson is a little bit more advanced form of matrices. He explains 2x2 matrices in another lesson. This lesson really helped me out in explaining the meaning of solving matrices. Thank you!

Nopic_grn
what
01/03/2010
~ Janet2

this video length is about 8 minutes, and it stops about halfway through so I don't get to see all of it, which doesn't help me at all.

Nopic_tan
Thank You!
10/17/2009
~ Assessment

I am a phd student. This little short video did more for me in 30 minutes or less than any review I have attempted thus far in trying to remember how to find the Inverse of a Matrix! Excellent! Good job!

Nachan_homepage
Matrices are easy!
01/16/2009
~ nachan

Professor Burger teaches all the aspect of matrices. This particular lesson is a little bit more advanced form of matrices. He explains 2x2 matrices in another lesson. This lesson really helped me out in explaining the meaning of solving matrices. Thank you!

Now I’m going to show you how you can adapt the previous method we just talked about, that method of sort of taking little teeny determinants and making a big matrix. Then taking a flip along the diagonal, and then dividing that by the scalar of 1 over the determinant to actually get the inverse. It’s a big old mess, but it turns out it actually works.
Let’s take a look at an example together. Suppose the matrix, A, which in fact is a three by three matrix, looks like this, 1, 0, 1, 2, -2, -1, 3, 0, 0. And my goal is to find the inverse of this matrix. What is that? That’s going to be another three by three matrix that has the property, if I matrix multiply the inverse by this, I should get the identity. That’s the matrix that has ones a long the diagonal, and zeros everywhere else.
So how do I proceed? Well the first thing I actually have to do is find the determinant of this matrix, because I have to have 1 over the determinant way out in front of my new matrix. So let’s just make a little side calculation. So first find the determinant. So what’s the determinant? Well let’s give it a shot, shall we? I’ll expand around here. And remember how this goes? Finding the determinant, I take this term and I write it down. I multiply that by the determinant of what I have left. But notice the determinant of that is 0, because it’s 0 - 0. So in fact that’s easy, that’s good, that’s 0. Then remember the signs. I’ve got a plus minus here, so I take a minus 0 times, well this determinant. Well that determinant turns out to be actually 3, but since this is a 0, who cares. But I’ll write it in anyway. And then plus, minus, plus. So this would be a +1 times, and what’s the determinant when this is all taken away? Well you can see this is going to be a 0 and then minus -6, so it’ll just become 6. So in fact, I see that the determinant of this matrix is 6. So the determinant of this matrix, I’ll write that down, we computed to be 6.
So the inverse, we’re going to start off by dividing by 6. So we’ll have a scalar in front. Okay, that’s just a little calculation that we’ll need for later. Now how do I proceed?
Well now I’m going to make this new matrix, which I call N. And it’s going to be sort of made in a similar way as we made, as we computed the determinant, but this is going to be a little teeny bit different. So remember how this goes. If I want to find out what goes in this spot, I go to the analogous spot here, and I block that off, and I see the matrix that’s left, and take its determinant and write that number right in here. I don’t multiply by that value, like you do when finding the determinant of the whole matrix like I just did. This is now something special. I’m building a new matrix. And the entry here is just the determinant of this thing right there. So I put that in. And what do I see? Well I see a 0, 0 so I just see 0. Now to find this entry, I just go to that analogous entry there, block off those people, and take the determinant. And put that answer in here. Here I see a 0 and a minus -3, which is a 3. I do the same thing here. I come here, block off the analogous spot here, see what’s left, take the determinant. This would be 0 minus -6, which would be 6. But now I’ve got to keep going. To find out what spot goes here, I just block the analogous spot off here, which would be this, see what’s left, and take the determinant. In this case, I’d see 0, 0, so I’d see 0. What about this middle one? It’s sort of harder to do, but there it is. The middle one would be a 0 - 3, and that last one, I just block that analogous spot off here, and I’d see a 0, 0. See how I’m making a new three by three matrix? And all I’m doing is I’m taking these little minor determinants. In fact, these are actually called minors, but who cares. What about what goes here? What do I do? I just block off the analogous thing here, take a determinant. This would be a 0 minus -2, which is 2. What about here? I just block off that spot, analogous spot. I see -1, -2 is -3. And then what about that last spot? I block off that last spot, and I see -2 + 0. So I see -2. So there’s the new matrix. However, remember I’ve got to put in the sign chart thing, like we did before, so I’ve got to put in a plus, minus, plus, and then I do the opposite here. So minus, plus, so this doesn’t change, minus. And then here I put in a plus, minus, and then a plus. So notice that some of the signs do change. And what I see now is the following. My new matrix, if I write this out nice and neat, looks like this. I see 0. I see -3. I see 6. I see 0, -3, 0, 2 and then a minus -3, is actually just 3, -2. So some of those signs actually changed when I put in the plus, minus, plus, minus, plus, minus, and so on.
Okay, now that’s the new matrix. To find the inverse, here’s what you do. So A inverse, so the inverse of this original matrix, is 1 over the determinant. So you take 1 over the determinant, which we computed earlier. So that’s . And then here what you do is you take the new matrix we have, and you just flip it over this line. So let’s do that. The diagonal elements will actually stay the same. So I’ll just have this 0, -3, and -2. But then this term and this term flip. So I put the 2 way up here, and I put the 6 way down here. And then these two terms flip. So I have a -3 and a 0 here. And these two terms flip. And I see a 3 and a 0 here. Whew! Well it turns out that that actually is the inverse. And you can multiply everything through by , which means that I would just see 0, 0, , , , , , 0 and -2. So in fact, I guess I could write that like this. I could say the inverse equals 0, 0, and is just another way of saying . And is just another way of saying , and then I see another . And then I see just . And here I see a 1. Here I see a 0. And here I see a . And you can check this. It turns out if you take this matrix and you multiply it by this matrix, what you would see everywhere is going to be the diagonal identity matrix. If you multiply these two things together, you’ll just see 1, 1, 1 and then zeros everywhere else. You’ll see the identity. And you can try that if you want.
The method though was a little bit elaborate, but actually it’s about as easy as it gets. You first find the determinant very carefully, and then you find this new matrix, which is a little bit tricky. To find the new matrix, what you do is you first of all block off people and just see what those minor determinants are, and write those minor determinants in the appropriate spots. You then put the plus minus grid in and change signs where appropriate. Then flip along this diagonal. Put that matrix here, and then in front, scalar multiply it by the reciprocal of the determinant, and you’ve got the inverse. And in fact this, by the way, works no matter how big of a dimension you have. So if you have a four by four matrix, you can actually now find the inverse of that. But boy I hope that no one would be so mean as to actually have you do that. By the way, one last comment, a really easy way of finding determinants, even easier than this, is to use a computer or a calculator. They actually give them to you for free. Anyway, try these. Work through them carefully and slowly. You can get them. You can.

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