Expert: Sherry Wallin - 4/7/2009

Question
Hello,

I'm a 25 year old computer science student. I'm not sure if linear algebra is considered abstract or not, so please excuse me if I made a mistake.

There's a question regarding determinants that I've been working on for a whole day and haven't been able to solve.

Given a square matrix "A" n*n containing only "ones" and "minus ones", I need to prove that exists an integer K so:
det(A) = K*(2^(n-1))

I've tried induction and thought I came close but reached a dead end after extracting 2^(n-1) from the summary. I needed to prove that whatever was left in the summary after the extraction - is even. So maybe induction is not the way.

Thanks,
Or

Answer
Hi Or~

Some thoughts:
If you take any nxn matrix you can find it's determinant by using cofactors that are 2x2 matrices. Examine all the cases of 2x2 matrices of 1's and -1's (there are 8 and in fact only three values come out and that is 0, 2, -2). So this means each cofactor can only be 0, 2, -2 and the sum or product of even numbers is even. This is a valid proof just fill in all the details. Write me again if you need more clarification.

Math Prof