07-03-2005 07:58									#1
	is there a very easy to code way of calculating the determinant using the inverse? Increase the size of your Attachments

I've painstakingly debugged my code that brings a matrix into reduced echelon form, to calculate the inverse. The prospect of writing the recursive function for the determinant is not inviting. Isn't there some way of using the inverse to find the determinant?

	07-03-2005 08:07									#2
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a) the determinant of the inverse is the inverse of the determinant

b) instead of using a recursive function to calculate the determinant by minors, use the the even/odd permutation definition of determinant

c) write down the eigenvalue equation for the matrix, solve for the roots and multiply them together to get the determinant

	07-03-2005 08:13									#3
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The determinant is properly defined as:

the sum over all permutations sigma of (-1)^(sign of sigma) a1sigma(1)a2sigma(2)...ansigma(n)

	07-03-2005 08:14									#4
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quote: I've painstakingly debugged my code that brings a matrix into reduced echelon form, to calculate the inverse. The prospect of writing the recursive function for the determinant is not inviting. Isn't there some way of using the inverse to find the determinant?

Not unless its inverse is easy to calculate.

Easiest way would be this:

detA = e_q1...qnA₁^q1...A_n^qn

Edit: In case it's not clear, the summation over each qm is implied through repeated indices. And e_q1...qn gives 1 if q1..qn is an even permutation of 1...n, -1 if it's an odd permutation.

	07-03-2005 08:15									#5
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MATH DORK THREAD!@@!~!!@!!!!@111one!!11!!!

	07-03-2005 08:23									#6
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If you've written down the row-reduced echelon form of the matrix and this has non zero determinant (i.e. it's the unit matrix) then if you've kept track of the operations performed to get ther then the determinant is trivial to calculate...

	07-03-2005 08:24									#7
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quote: Originally posted by KrazyHorse If you've written down the row-reduced echelon form of the matrix and this has non zero determinant (i.e. it's the unit matrix) then if you've kept track of the operations performed to get ther then the determinant is trivial to calculate...

*looks at wikipedia*

Thanks! That's really, really easy to implement.

	07-03-2005 08:26									#8
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How big is this matrix?

	07-03-2005 08:39									#9
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Are you writing this in Fortran?

	07-03-2005 08:42									#10
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quote: Originally posted by KrazyHorse How big is this matrix?

Arbitrarily sized (= I hardcoded allocation of 100x100 ints assuming my instructor won't wan't to type 10,000 numbers in to check my program ).

And it's in C.

	07-03-2005 08:51									#11
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Won't it take forever to run anything larger than 20X20 or so?

	07-03-2005 08:51									#12
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Probably.

(The assignment is a linear algebra assignment, it's more my computer architecture class. It's more a "can you write in C, and can you do all this stuff while handling all sorts of crap user input and make it bug-free" sort of assignment.)

	07-03-2005 08:52									#13
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I think exact inversion runs in N^3 time. Can't remember.

	07-03-2005 08:53									#14
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My algorithm is N^3, I think.

	07-03-2005 08:54									#15
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approximate numerical inversion is much faster.

	08-03-2005 06:57									#16
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n being number of entries of number of lines?