trace and determinant from eigenvalues
Two of the most basic numbers attached to a matrix — its trace (sum of diagonal entries) and its determinant — turn out to be completely determined by the eigenvalues. The trace equals the sum of the eigenvalues, and the determinant equals their product, each counted with algebraic multiplicity. This ties a coordinate-bound computation (the diagonal) to a coordinate-free invariant (the spectrum).
Over C, where the characteristic polynomial p(lambda) = det(lambda I - A) splits as (lambda - lambda_1) ... (lambda - lambda_n), comparing coefficients gives the two identities directly: tr(A) = lambda_1 + ... + lambda_n and det(A) = lambda_1 ... lambda_n. They are the first and last of the general elementary-symmetric relations between coefficients of p and the eigenvalues.
These are wonderful sanity checks. After you compute eigenvalues, verify their sum equals the trace and their product equals the determinant; a mismatch means an arithmetic slip. They also explain instantly why a matrix is invertible iff it has no zero eigenvalue (det != 0), and they are invariant under similarity, since similar matrices share both trace and determinant.
Trace is the sum of eigenvalues; determinant is their product — both counted with multiplicity.
More generally, the coefficient of lambda^{n-k} in the characteristic polynomial is (up to sign) the k-th elementary symmetric polynomial in the eigenvalues, equal to the sum of the k-by-k principal minors of A. Trace (k=1) and determinant (k=n) are just the endpoints.