Linear Algebra scholarship exam

[Maybe_Memorie]

Homework Statement

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

Homework Equations

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.

 

[micromass]

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

 

[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

 

[micromass]

so det(A) =1

or -1.

That is indeed the correct reasoning!

 

[Maybe_Memorie]

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