  We remind the reader that it is possible to tell if two or more eigenvalues
of a matrix are equal, without calculating the eigenvalues. We then use this
property to detect (avoided) crossings in the spectra of quantum Hamiltonians
representable by matrices. This approach provides a pedagogical introduction to
(avoided) crossings, is capable of handling realistic Hamiltonians
analytically, and offers a way to visualize crossings which is sometimes
superior to that provided by the spectrum. We illustrate the method using the
Breit-Rabi Hamiltonian to describe the hyperfine-Zeeman structure of the ground
state hydrogen atom in a uniform magnetic field.
