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College Algebra: Evaluating nxn Determinants

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:00
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 97 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: Evaluating Determinants (3 lessons, $4.95)

With larger square matrices, the calculation of the determinant gets more difficult. This lesson shows you a special method to use to identify the determinant of a 3X3 square matrix. You will also learn another technique to use to calculate the determinant of a 3X3 or larger square matrix. Professor Burger will go over the rules for identifying the determinant of any square matrix.

For Professor Burger's lesson on evaluating determinants of 2X2 sized matrices, check out http://www.mindbites.com/lesson/804.

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
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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/.

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So as your square matrices get larger and larger it turns out that computing the determinant for that matrix is going to get trickier and trickier as you might have guessed. In the two by two cases it’s really easy. You can sort take that cross set of products there and subtract them and you actually get the determinant. But what about for a three by three? For instance, what about this matrix right here? Well, there’s actually for a three by three there’s two methods. You can actually sort of adopt the previous method of those diagonal things, if you’re careful. I’ll just say this in words so you can just hear it once, but then I’ll show you a way that actually can be used no matter what size the matrix is.
So, with a three by three what you could do is try to adopt the method used for two by two. Remember two by two was take this and then subtract it from this. Well, you can sort of do that here, but here is a lot of diagonals and you have to consider all of them. So, let me just show you what you would do. What you would do is you would take this times this times this. You would take this diagonal of three then you would actually add the other diagonals of three that are going in this direction. For example, this, this and then you have to come back and do this one. See, one, two and then three. That would be the other diagonal of three in this direction, and then you’d have zero times one and three. There’s another diagonal of three in this direction.
So, you’d have this diagonal of three, this diagonal three and this diagonal of three. You’d take the products of each of those three numbers individually, add them up and then subtract from that all the diagonals of three in the other directions. Which would be this diagonal, this diagonal and this diagonal. So, in fact, you could find all the diagonals of three that go this way. So, one, two, three and subtract from that one, two three. That would work. But unfortunately it doesn’t generalize the higher and higher sizes of matrices. So, let me show you a way that actually will always work and it’s sort of the divide and conquer method. That is, once you know how to find determinants of two by two matrices let’s use that fact to find determinants of larger matrices.
So, the first thing you have to remember is to make a little sign chart. It’s easy to make the sign chart. You just put in a sign for each position of the matrix and you start with a plus and you alternate. So, plus, minus, plus, minus, plus, minus, plus, minus, plus. So, there’s the little sign chart. It’s always the same thing. You start with a plus and alternate as you go through. So, you get sort of like a checkerboard looking like thing here. Now, here’s how you find the determinant of a three by three matrix. You pick any row at all. So, it could be this row or this row or this row and that’s the row that you want to sort of expand about.
What you do--let’s suppose we just pick this row here to expand about. I take this number right here, two, and I’m going to multiply it by what I would get if I remove the column and the row that contains two. Let’s think about that. The column that contains two is right here. So, let’s get rid of that one. Now, let’s get rid of the row that contains two. That’s this one. If I do that, now look what I’m left with. I’m left with a two by two matrix. I can take its determinant pretty easily. It’s just this cross thing. So, I take that determinant and I multiply it by that two. So, I take two times the determinant I get if I cross out the things that contain two.
So, basically what I’m doing is this. I’m taking the two and I’m crossing out the column and the row that contains two. I’m taking the determinant of this and multiplying it by two. That’s the first term I’m going to have in my determinant. Now, I take that and I add that because I see a plus sign here. Now, I jump over to the next element in this row, which is a one, and I do the same trick. Namely, I block out the column and the row that contains one. If I do that you’ll notice now I still have a two by two matrix left. You have to sort of push them together now, but you can still see it’s three, four, one, two. I can take the determinant of that and I can multiply it by one. Then what I’ll do, since is see a negative sign here, is I put a negative in front of that.
So, I take the previous number I got and subtract this because that’s what the sign chart says. Then finally I go to this last element and use the same procedure. I block out the row and the column that contains it. I see a little two by two matrix left. I can compute it’s determinant. Take its determinant and multiply it by this and then I, in this case, add it. So, the procedure is one where you sort of take a number and multiply it by the determinant of what’s left over. In this case I would subtract, because the signs alternate, minus and take that number and multiply it by the determinant you get of what’s left over. Then, finally, I add to it this number times the determinant that’s left over when you block that out.
So, let’s try that and actually see what you would get. So, in this case, the determinant--let me call this matrix for example A, just for to give it a name. So, the determinant of A would equal what? Well, the first thing I’ll do is I’ll expand around this row. So, I say, okay it’s going to be two time, so it’s going to equal two times--there’s the two and now I have to take the determinant of the two by two matrix I get by getting rid of the column and the row that contains two. I just block them out. The little two by two matrix there, what’s its determinant? It’s -2 -4. Because remember, it’s -2 - 4. So, -2 - 4 is -6.
Then on my sign chart I see a negative sign, which means the next determinant is going to be subtracted. So, you alternate. You add, you subtract, you add and so forth. So, now I subtract and then I do the same thing. I expand around that middle element. Which means I take one and I multiply it by the determinant of what I get if I remove the row and the column that contains one. I remove that T and you can see a two by two matrix here. What’s the determinant? 3 x 2, which is 6, minus 4 x 1, which is 4. So, 6 - 4 is just 2. So, I put a two there. I’m subtracting because the signs alternate. The way I combined them alternate.
Then finally, I take and this is sort of the fun one, I expand around this zero point, which means I get rid of the row and the column that contains zero and I take zero time the determinant of this. Well, that’s going to be easy, because no matter what the determinant is, since I’m multiplying by zero it’s going to be zero. So, you see, actually expanding around zero is really neat and I add so I have plus zero. I don’t have go any further, but just to show you what I would do, I’d say 3 x 1, which is 3 and this is a -1, so I’d have 3 - (-1) gives me 4. It doesn’t matter. I’m just trying to show you how I take the determinant of this.
Therefore, the determinant of the whole matrix is going to be -12 - 2 which equals -14. So, the determinant of this is -14. I found it by just expanding along this particular row. Now you could have expanded along this row if you wanted to too, the exact same way. Take three and multiply it by the determinant you get by doing that. That’s a negative sign there, so you’ve got to now realize that you start by subtracting. The next thing is a positive, so you take negative one and you crisscross that out. Can you see the matrix there? It’s two, zero, one, two. You take the determinant and add that and then you can expand around the four, but you’ve got to subtract it because the sign is here. Four times and then what you get here, which would be two, one, one, one.
So, you can expand around any row. In fact, you can even expand around any column you want. You just have to keep in track of this sign chart which just tells you how to add and subtract the numbers together in order to find the determinant. This also works for four by fours. With four by fours you can do the same thing, but then of course, when you start to block things off what you’re left with is a three by three. So, that requires all the three by three methods. You can see it gets harder and harder, but now you can see how to find the determinant of anything at all.
Now, the thing I want to tell you about next is why should I care about finding determinants. I’m going to show you a really cool little just fun application just for us. Up next to show you that the determinant actually is pretty cool and then later we’ll look at some real serious applications of the determinant. Until then, just try to work through these examples and remember what you’re doing. When you’re expanding around a particular row you take the element here, with the appropriate sign and multiply it by the determinant of the matrix that remains after you get rid of column and row. Take determinant, add or subtract them, you’ve got it made. Enjoy.

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