What is the formula for finding the determinant of a 10x10 matrix?

[Zythyr]

I am trying to figure out the formula for the determinant of a 10x10 matrix. I was told to use row reducation method, but I am not really sure what it is. I never took Linear Algebra. Can someone please help me.

 

[gendou2]

This website has a nice explanation of the method:

http://tutorial.math.lamar.edu/Classes/LinAlg/DeterminantByRowReduction.aspx

 

[Defennder]

It's going to be quite tedious even if you were to row-reduce it before finding its determinant. Some special matrices have easy determinants, so maybe you could see if the matrix for which you are trying to evaluate the determinant has some property which allows you to compute its determinant easily.

EDIT: I see that you say you are trying to "figure out the formula". There's a recursive method for finding the determinants of an arbitrary nxn matrix. It's known as cofactor expansion.

 

[Borek]

I know that's not what you are asking for, but in the case of 10x10... go numerical.

 

[BryanP]

Definitely go numerical. It's going to take a long time computing the determinant of that unless it was diagonal.

 

[Zythyr]

Umm I can't go numerical... I need to do it in terms of formulas... And no, diagonals arent allowed. What can I do? Can someone help me determine the forumla? please

 

[robphy]

gendou2 and Defennder gave you a general method

"Cofactor expansion" or "Laplace expansion"
http://tutorial.math.lamar.edu/Classes/LinAlg/MethodOfCofactors.aspx (has 5x5 example)
http://en.wikipedia.org/wiki/Expansion_by_minors

 

[maze]

Put the matrix in a list and use guassian elimination, and sounds like the perfect sort of problem for a functional programming language.

 

[Haelfix]

The explicit formula for such a matrix will be a horrible, horrible mess (think many many pages).

You are almost guarenteed to make an algebra error somewhere. This is exactly what computers are for.

 

[Defennder]

Umm I can't go numerical... I need to do it in terms of formulas... And no, diagonals arent allowed. What can I do? Can someone help me determine the forumla? please

Is this question from a textbook? If so, then perhaps it's best if you were to post the exact problem. The problem with devising a formula for the determinant of a 10x10 matrix is that it would require far too many variables, at least 100 variables would be needed, each for every entry of the matrix. I doubt any textbook problem would require such to be done.

 

[Zythyr]

Is this question from a textbook? If so, then perhaps it's best if you were to post the exact problem. The problem with devising a formula for the determinant of a 10x10 matrix is that it would require far too many variables, at least 100 variables would be needed, each for every entry of the matrix. I doubt any textbook problem would require such to be done.

Not in a textbook. My proffesor for Diffiq said if anyone figures out the forumal for the dertminant of a 10x10, they automatically get an A in the class.

 

[Defennder]

I'm pretty sure he meant that tongue-in-cheek. To clarify, did you ask him why he wanted only the formula for a 10x10 matrix and not some other arbitrary size?

 

[robphy]

If you really want it [and don't want to derive it], you can write a short Maple program.

with(LinearAlgebra); M:=Matrix(3,3,symbol=m); Determinant(M);

You might wish gradually tune the size of the square matrix up to your desired value... but you should be prepared to wait.

 

[LukeD]

Finding the formula is really simple. You'd just need several pages to write/print it, and there would be 100 variables. So you'd be very likely to make a mistake somewhere if you tried to do it by hand.

I doubt that your professor would actually give an A for it though since it is extremely easy.

Hell, I wonder if this would suffice: (Let $$a_{i,j}$$ denote the i,jth entry of the matrix)
$$\sum_{\sigma \in S_{10}} \text{sgn}(\sigma) \sum_{i=1}^{10} a_{i, \sigma(i)}$$
because that is one way to write the formula. It's called Leibniz's formula for the determinant. Of course you'd need to know what $$S_{10}$$ is and what the sign of an element of $$S_{10}$$ means as well as how to interpret the summation signs