Well, every time, we can make so-called "guess" that solution looks like e*rt. Why? We know that because professors will only give well-behaved systems on final exams because it's hard to grade the other kind.

So we know characteristic polynomials look like so (because of course they do, you can just memorize this) ... so now we lift out the coefficients into nifty thing called _matrix_ and now follow these easy four steps to get roots, plug back in, and incidentally these are "eigenvalues", we'll talk about this later ...

Bam. Done. A-, easy. No sweat."