Matrix mechanics represents an observable by an array of matrix elements between chosen quantum states. In an orthonormal basis of energy eigenstates, each a stationary state, the diagonal entries of the Hamiltonian operator are the energy levels; off-diagonal entries of other observables describe couplings between those states. The same array can therefore answer two different questions: which values can a measurement return, and which pairs of energy eigenstates can a specified interaction connect? The eigenvalues answer the first question; the relevant matrix elements, together with the energy differences, answer the second.
This makes linear algebra a practical language for atomic physics. One can calculate energy levels, electric-dipole selection rules and probability amplitudes for transitions without assigning an Electron an unobserved classical orbit. The three-state example below separates a missing optical coupling from a missing energy level; the finite oscillator commutator boundary defect explains why a finite array requires care when approximating a quantum harmonic oscillator.
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Published by Werner Heisenberg in 1925-07-25 as quantum mechanical re-interpretation of kinematic and mechanical relations by Heisenberg (1925), it offered the first general formulation of quantum mechanics.
It is apparently more closely related to the ladder operator method, which is a more algebraic than the more analytical Schrödinger equation.
It appears that this formulation makes the importance of the Poisson bracket clear, and explains why physicists are so obsessed with talking about position and momentum space. This point of view also apparently makes it clearer that quantum mechanics can be seen as a generalization of classical mechanics through the Hamiltonian.
QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) mentions however that relativistic quantum mechanics broke that analogy, because some 2x2 matrix had a different form, TODO find that again.
Inward Bound by Abraham Pais (1988) chapter 12 "Quantum mechanics, an essay" part (c) "A chronology" has some ultra brief, but worthwhile mentions of matrix mechanics and the commutator.