In general, given a $3\times 3$ or $4\times 4$ matrix $A$ which doesn't have a lot of $0$ entries, what is the fastest or less error prone way to compute its minimal polynomial?
More generally, I have a differential equations final coming, and this is required for solving $x' = A x$ problems (where $A$ is a matrix). I usually fail to solve those because there is a lot of computing involved and I make silly errors which later turns into horrors (integrals, systems...).
I am worried about making these silly errors and fail the exam even when I know the theory and the methods.
Do you have any tricks to solve these kind of standard problems, where $A$ is usually a $3\times 3$ matrix with integer entries?
I believe in solving lots of problems to get good, here: at solving first order matrix differential equations. I used the Schaum's Outline on Differential Equations to train, maybe your library has a copy. Mine is from 1988.
To calculate the characteristic polynomial, you need to be able to calculate determinants,
$$ \chi(\lambda) = \det \left( A - \lambda I \right) $$
I used the Schaum's Outline on Linear Algebra for that. :-)
Addendum: I removed the excerpt pages on $e^{At}$ calculation, as you say you know that part.