Max-plus algebra is the study of the max-plus semiring. Tropical mathematics is the more general field that concerns any semiring where the 'addition' operation is max or min. Tropical mathematics has numerous applications in dynamical systems and algebraic geometry. In this talk I will discuss my work that uses max-plus algebra in numerical linear algebra. The idea is that a conventional NLA problem can be transformed into a max-plus one, which is typically easier to solve. The solution to the max-plus problem reveals important structure in the original problem, which can be exploited to help in its subsequent solution. My talk will cover techniques for choosing the sparsity patterns for incomplete LU preconditioners as well as choosing the importance sampling distributions to be used in matrix sketching algorithms.