Matrix multiplcation plays an important role in quantum mechanics, and all throughout physics. Examples include the moment of inertia tensor, continuous-time descriptions of the evolution of physical systems using Hamiltonians (especially in systems with a finite number of basis states), and the most general formulation of the Lorentz transformation from special relativity.
General relativity also makes use of tensors, which are a generalization of the sorts of objects which row-vectors, column-vectors, and matrices all are. Very roughly speaking, row- and column-vectors are 'one dimensional' tensors, having only one index for its coefficients, and matrices are 'two dimensional' tensors, having two indices for its coefficients, of two different 'kinds' representing rows and columns — input and output, if you prefer. Tensors allow three or more indices, and to allow more than one index to have the same 'kind'.