Rational Canonical Form
You may look up the title of this post if you wish, but it isn't important. I had the following on an algebra homework:
Find a matrix A with rational entries such that A is not the identity matrix and A^3 = I.
Now I knew I was supposed to use the rational canonical form, but being lazy, the question seemed easy enough to solve by just multiplying everything out. However, I quickly found this is not the case.
What I wish to see is that if anyone can solve the question using a reasoned method. In other words, guess and check is not allowed.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."