# Why are self-adjoint operators important?

Addressing your second question first, finding exact eigenvalues becomes quite difficult and computationally intensive as the size of the matrices involved increases. In particular, there is (provably!) no general formula for finding the roots of a polynomial of degree 5 or higher, so we might not even be able to explicitly find the eigenvalues for a matrix $$5 \times 5$$ or larger! Finding efficient numerical algorithms for approximating eigenvalues of large matrices is a huge field---for example, the original implementation of Google's search algorithm is built around approximating the dominant eigenvalue and eigenvector for the adjacency matrix of the entire internet.
As for why someone might care about self-adjoint operators, they take a pivotal role in physics and especially quantum mechanics. Given a physical system, all of the observables that we might ask (position, momentum, etc) are represented by self-adjoint operators in some appropriate context. For example, Heisenberg's uncertainty principle is often stated as $$PQ - QP = i\hbar$$ where P and Q are the (self-adjoint!) position and momentum operators.