Linear Algebra scholarship exam

1. Dec 7, 2011

Maybe_Memorie

1. The problem statement, all variables and given/known data

This is one of the three linear algebra scholarship questions given by my university last year. I've solved the other two, but this one is posing a bit of a problem. Question 1 on the file.

Given that for a nxn matrix A both matricies A and A-1 have integer entries, show that det(A) = +-1

2. Relevant equations

3. The attempt at a solution

I'm completely lost. I have a feeling co-factor expansion isn't the way to go, as that would be very messy, and the other two worked out fairly nicely when you know what you're doing.

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2. Dec 7, 2011

micromass

Can you show that det(A) and det(A-1) are integers??

3. Dec 7, 2011

Maybe_Memorie

I can reason it now, but not really mathematically show it.
All the entries of A are integers. So by cofactor expansion for det(A) along the first row every part will be a product of integers, so an integer. Same reasoning for det(A-1)

But det(A-1) = det(A)-1

So, an integer = 1/that integer, so det(A) =1

4. Dec 7, 2011

micromass

or -1.

That is indeed the correct reasoning!!

5. Dec 7, 2011

Maybe_Memorie

I can't edit for some reason, but I meant +-1