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Intuitive explanation for the general determinant formula?

  1. Dec 18, 2012 #1
    Hello

    Could anyone give an intuitive explanation of the determinant? I know mostly what the determinant means and I can calculate it etc. But I have never got any real-world intuitive explanation of the general formula of the determinant?

    How is the formula derived? Where does it come from? What I'm essentially asking is: Prove the general formula for calculating the determinant of an n-by-n matrix and explain the meaning of it

    Any support = Thank you so much!! =)
     
  2. jcsd
  3. Dec 18, 2012 #2

    jgens

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    Often the determinant is just defined by the formula:
    [tex]\det(a_{ij}) = \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{1\sigma(1)} \cdots a_{n\sigma(n)}[/tex]
    On the other hand, if you define the determinant as the unique alternating multilinear functional [itex]\det:\mathbb{R}^n \times \cdots \times \mathbb{R}^n \rightarrow \mathbb{R}[/itex] (where the product is taken n-times) satisfying [itex]\det(I) = 1[/itex], then you can recover the formula above for the determinant.

    Edit: I suppose this is not really an intuitive explanation, but hopefully it helps a little.
     
    Last edited: Dec 18, 2012
  4. Dec 18, 2012 #3

    micromass

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    A nice and intuitive explanation of the determinant is that it just represents a signed volume.

    For example, given the vectors (a,b) and (c,d) in [itex]\mathbb{R}^2[/itex]. Then we can look at the parallelogram formed by (0,0), (a,b), (c,d) and (a+c,b+d). The area of this parallellogram is given by the absolute value of

    [tex]det \left(\begin{array}{cc} a & b\\ c & d \end{array}\right)[/tex]

    Of course the determinant has a sign as well. This is why we call the determinant the signed volume. That is: if we exchange (a,b) and (c,d), then we get the opposite area. The sign is useful for determining orientation.
     
  5. Dec 18, 2012 #4

    tiny-tim

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    welcome to pf!

    hello jjepsuomi! welcome to pf! :smile:

    i suggest you start in 2D and 3D by considering how the determinant relates the the area of a rectangle to the area of the transformed parallelogram or the volume of a cube to the volume of the transformed parallelepiped,

    and then how you'd apply that in n dimensions, and how it affects integration after a transformation :wink:
     
  6. Dec 19, 2012 #5

    mathwonk

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    saying it is an oriented volume measure implies it should be a multilinear and alternating function. these properties force the formula to be what it is, if you assume the value is 1 on a unit cube. determinants are developed in complete detail starting on p. 62 of these notes.

    http://www.math.uga.edu/~roy/4050sum08.pdf


    for geometric intuition you might look at the book by david bachmann on geometric approach to differential forms.

    https://www.amazon.com/Geometric-Ap...1&keywords=david+bachmann,+differential+forms
     
    Last edited by a moderator: May 6, 2017
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