# Determinant of 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.

## Answers and Replies

Homework Helper
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.

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

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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

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

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.

Homework Helper
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.

Homework Helper
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?

Homework Helper
Gold Member
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