The algebra of matrices is not commutative, namely different operators don't commute each other, it is a fundamental characteristic of Quantum Mechanics.
In Quantum Mechanics the non-commutativity of the algebra of the operators, is expressed by the introduction of the commutator operator. Namely if I have two operators, represented by the matrices A, B, then the commutator operator is defined as follows:
[A,B] = A*B - B*A, which, of course is not equal to zero, in general. For example the commuattor between the operators coordinate and momentum is:
[x, p] = i* h/(2*pi), where i is such that i^2=-1 and h is the Planck constant, and pi = 3.14159...