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Intuitive explanation for the general determinant formula?

by jjepsuomi
Tags: determinant, linear algebra
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Dec18-12, 10:06 AM
P: 3

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!! =)
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Dec18-12, 01:25 PM
P: 1,622
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.
Dec18-12, 01:36 PM
micromass's Avatar
P: 18,333
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.

Dec18-12, 01:36 PM
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tiny-tim's Avatar
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Intuitive explanation for the general determinant formula?

hello jjepsuomi! welcome to pf!

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
Dec19-12, 12:26 PM
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mathwonk's Avatar
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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.

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

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