For a randomly generated $n$ by $n$ matrix, is the probability that it has $n$ distinct eigenvalues equal to $1$? I have a feeling it must be.
But, if that's the case, why do we concern ourselves so much with situations where such a matrix has repeated eigenvalues? Is it because such situations arise so often in the real world? And, if so, what is it about real-world problems that leads to such "rare" matrices?
Sorry if my question seems a bit vague, I just thought of it when looking over geometric/algebraic multiplicity.