Determinant of a MatrixDate: 11/05/97 at 20:38:18 From: Mark Geraghty Subject: Definition of DETERMINANT My Alegebra 2 teacher told us that for extra credit we could give him a complete and unabridged version of the definition of a DETERMINANT of a matrix. He said that we could only find it in a library or an advanced collegiate math book. He also said that if we could understand the definition, it wasn't the right one. I have tried numerous dictionaries and other references over the Internet. Please help me if you can! Thank you so much. Date: 11/06/97 at 14:30:31 From: Doctor Tom Subject: Re: Definition of DETERMINANT Hi Mark, I'll give you two definitions that are exactly equivalent, but sound very different. The first is geometric. I assume you've plotted things in an x-y coordinate system, right? I assume you can imagine doing the same thing in three dimensions with an x-y-z coordinate system as well. In 2-D, when you talk about the point (2, 4), you can think of the "2" and "4" as directions to get from the origin to the point - "move 2 units in the x direction and 4 in the y direction." In a 3-D system, the same idea holds - (1, 3, 7) means start at the origin (0,0,0), go 1 unit in the x direction, 3 in the y direction, and 7 in the z direction. Similarly, you could have coordinates in one dimension, but there's just one number. The determinant of a 1x1 matrix is the signed length of the line from the origin to the point. It's positive if the point is in the positive x direction, negative if in the other direction. In 2-D, look at the matrix as two 2-dimensional points on the plane, and complete the parallelogram that includes those two points and the origin. The (signed) area of this parallelogram is the determinant. If you sweep clockwise from the first to the second, the determinant is negative; otherwise, positive. In 3-D, look at the matrix as 3 3-dimensional points in space. Complete the parallepiped that includes these points and the origin, and the determinant is the (signed) volume of the parallelepiped. The same idea works in any number of dimensions. The determinant is just the (signed) volume of the n-dimensional parallelepiped. Notice that length, area, volume are the "volumes" in 1-, 2-, and 3-dimensional spaces. A similar concept of volume exists for Euclidean space of any dimensionality. Okay. That's the geometric definition. I like it because I can make a mental picture of it. Here's the algebraic definition: I'll do it in 3 dimensions, but exactly the same idea works in any number of dimensions. Let's look at the determinant of this matrix: | a11 a12 a13 | | a21 a22 a23 | | a31 a32 a33 | The numbers after the "a" are the row and column numbers. A permutation of a set of numbers is a re-arrangement. For example, there are 6 permutations of the list (1 2 3), including the "re-arrangement" that leaves everything unchanged). Ignore for the moment the "+1" and "-1" after each one: (1 2 3) -> (1 2 3) +1 (1 2 3) -> (1 3 2) -1 (1 2 3) -> (2 1 3) -1 (1 2 3) -> (2 3 1) +1 (1 2 3) -> (3 1 2) +1 (1 2 3) -> (3 2 1) -1 Now imagine that you start with three objects labelled 1, 2, and 3 arranged as they are on the left, and need to convert them to the order on the right, but you're only allowed to swap one pair at a time. To get to the final arrangement, you'll find that there are lots of ways to do it, but every way (for a particular rearrangement) always requires an even number of swaps or always requires an odd number of swaps. I've labelled those that always need an even number of swaps with +1 and those needing an odd number as -1 above. Now write down 6 products of the "a" terms, where the first number for each term is 1, 2, 3 and the second number is the rearrangement above for each of the six rearrangements. Here's what they are, in the same order as above. Be sure you understand this step: a11*a22*a33 a11*a23*a32 a12*a21*a33 a12*a23*a31 a13*a21*a32 a13*a22*a31 The determinant is just the sum of all 6 terms, but put a "+" in front if the rearrangement is even, and a "-" in front if the rearrangement required an odd number of swaps. Here's the answer: +a11*a22*a33 -a11*a23*a32 -a12*a21*a33 +a12*a23*a31 +a13*a21*a32 -a13*a22*a31 For a 4x4 matrix, there will be 24 rearrangments, like this: (1 2 3 4) -> (3 2 4 1) +1 ... so there will be 24 terms in the expression of the determinant. For a 5x5 matrix there are 120 rearrangements, so there will be 120 terms in the determinant, and so on. For an NxN matrix, there will be N! (N factorial) terms, where factorial means you multiply together all the terms from N down to 1. For example, 5! = "5 factorial" = 5x4x3x2x1 = 120. I hope this helps. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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