True of matrix multiplication, or a habit imported illegally from ordinary numbers?

True for matrices: $A(BC) = (AB)C$, $A(B + C) = AB + AC$, $AI = A$ for the identity $I$

False habit: $AB = BA$ always, $AB = 0$ forces $A = 0$ or $B = 0$, $AB = AC$ lets you cancel to $B = C$

Associativity and distributivity survive the trip from numbers to matrices; commutativity, zero-divisor innocence, and cancellation do not.