Matrix mechanics is the first way that physicists found to express quantum physics in mathematical form. Werner Heisenberg initially developed this form of expression of the laws of physics just as an equation that he could use to predict the intensities of the photons in the various bands of the hydrogen spectrum.
Heisenberg's teacher and colleague, Max Born, saw that his equation was essentially the plan for creating and multiplying matrices. The matrix form of quantum physics is still used since it is useful and convenient for some purposes. Other mathematical ways, particularly Erwin Schrödinger's equation using a wave function, are mathematically equivalent but are easier to use for other purposes.
One of the early successes of this theory was announced shortly thereafter and is now called the Heisenberg's Uncertainty Principle.