Matrices Formula for 10 by 10 Matrices

BionicBiceps

I'm looking for a determinant formula for a 10 by 10 matrices in variable format.

Robert1986

OK. Why do you want to know this? Do you have a specific example? This is going to be rough if the matrix isn't of a very simple type. For example, if it was sparse, it might be easier to figure out. Any general formula is going to involve 100 variables and is going to be an incredible mess and is going to be VERY VERY LONG. Basically, the first step is going to be finding the determinants of 10 9x9 matrices. This will be tough.

So, can you post a specific matrix for which you want the det?

BionicBiceps

I just need the basic formula for any 10x10 matrices

Robert1986

Yeah, I got that, but what I'm saying is that the "basic formula" is incredibly complicated. It will involve 100 variables. Surely you can see why that will be a mess. That, combined with the fact that the first step will be to find the det of 10 9x9 matricies. It will be long and will take up many pages.

Here is the det of a 5x5 matrix (with the first row being a,b,c,d,e and so on)

a g m s y-a g m t x-a g n r y+a g n t w+a g o r x-a g o s w-a h l s y+a h l t x+a h n q y-a h n t v-a h o q x+a h o s v+a i l r y-a i l t w-a i m q y+a i m t v+a i o q w-a i o r v-a j l r x+a j l s w+a j m q x-a j m s v-a j n q w+a j n r v-b f m s y+b f m t x+b f n r y-b f n t w-b f o r x+b f o s w+b h k s y-b h k t x-b h n p y+b h n t u+b h o p x-b h o s u-b i k r y+b i k t w+b i m p y-b i m t u-b i o p w+b i o r u+b j k r x-b j k s w-b j m p x+b j m s u+b j n p w-b j n r u+c f l s y-c f l t x-c f n q y+c f n t v+c f o q x-c f o s v-c g k s y+c g k t x+c g n p y-c g n t u-c g o p x+c g o s u+c i k q y-c i k t v-c i l p y+c i l t u+c i o p v-c i o q u-c j k q x+c j k s v+c j l p x-c j l s u-c j n p v+c j n q u-d f l r y+d f l t w+d f m q y-d f m t v-d f o q w+d f o r v+d g k r y-d g k t w-d g m p y+d g m t u+d g o p w-d g o r u-d h k q y+d h k t v+d h l p y-d h l t u-d h o p v+d h o q u+d j k q w-d j k r v-d j l p w+d j l r u+d j m p v-d j m q u+e f l r x-e f l s w-e f m q x+e f m s v+e f n q w-e f n r v-e g k r x+e g k s w+e g m p x-e g m s u-e g n p w+e g n r u+e h k q x-e h k s v-e h l p x+e h l s u+e h n p v-e h n q u-e i k q w+e i k r v+e i l p w-e i l r u-e i m p v+e i m q u

BionicBiceps

Is there a simpler method then row reduction for solving this?

Robert1986

Not that I am aware of. But, I ask again, WHY do you need to know this? Let me guess, your Diff. Eq. prof promised you an A if you could give him a formula for the det of a 10x10 matrix. Is that right?

BionicBiceps

haha that's exactly why

Robert1986

And now you can see why he told you this.

BionicBiceps

There has to be an easier way through a computer program.

Robert1986

Well, sure, but that is probably going to involve either the method I mentioned or some sort of numerical method. The only reason it is 'easier' is because the computer can do it much faster than you can (though doing a 10x10 matrix will probably take a REALLY long time.)

The Electrician

A formula of the form that Robert1986 has already given you for a 5x5 matrix, if derived for a 10x10 matrix, will consist of 10! = 3,628,800 terms, with each term a product of 10 variables. If printed out with 10 terms per line, 50 lines per page, it will take over 7000 pages; that's several reams of paper!