Intuitive explanation for the general determinant formula?by jjepsuomi Tags: determinant, linear algebra 

#1
Dec1812, 10:06 AM

P: 3

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 realworld 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 nbyn matrix and explain the meaning of it Any support = Thank you so much!! =) 



#2
Dec1812, 01:25 PM

P: 1,623

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 ntimes) 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. 



#3
Dec1812, 01:36 PM

Mentor
P: 16,565

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. 



#4
Dec1812, 01:36 PM

Sci Advisor
HW Helper
Thanks
P: 26,167

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 



#5
Dec1912, 12:26 PM

Sci Advisor
HW Helper
P: 9,421

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/%7Eroy/4050sum08.pdf for geometric intuition you might look at the book by david bachmann on geometric approach to differential forms. http://www.amazon.com/GeometricAppr...erential+forms 


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