Linear Algebra presentation Ok, so let me give you the background story. 
In my country, unlike the US, there are different high schools for different levels of 'intelligence'. I believe there are 5 or 6 types, the toughest being 'gymnasium' which is for the top 5% I believe. 
I am currently enrolled at a 'gymnasium', and they have a special program for the most gifted out of the gifted. I am in this program, and in this program we basically get a couple of hours a week off to work on a presentation for the entire school (in April every year). 
Most of the children participating in this program just do nothing in the hours they take off and just give a musical performance, but I wanted to do it about Linear Algebra this year, since I've started a self-study almost a week ago (now I'm learning about null spaces). (Sub-question by the way: Am I fast or not? I started L.A. 5 days ago, never having seen a matrix in my life and now I am learning about null spaces.. but it seems I am going too slow (or maybe that's just because I'm so excited and want to learn more and more every day)). 
My main question is: 
What would be interesting in a presentation of Linear Algebra? What concepts, ideas, etc. are mind-blowing and would be nice (for an intellectual crowd, don't worry about people being completely disinterested in math). 
You can give an advanced answer, it's October and I still have half a year!   

Probably not the most amazing, but fairly easy to explain to a non-nerd crowd:
finding area and volume using determinants.
