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If A is m × n and B is n × p, their product AB is m × p where (AB)_ij = Σ_k a_ik b_kj. Matrix multiplication is associative and distributes over addition but is NOT commutative. It represents function composition for linear transformations. Multiplication is possible only when inner dimensions match.
Think of matrix multiplication as a machine that transforms input vectors step by step. When you compute AB, you are asking: "first apply transformation B, then apply transformation A." The result AB is a single matrix encoding the composition of both transformations — just as composing two functions f(g(x)) gives a single combined function. This is the deepest reason matrix multiplication exists: it is function composition made computational.
The row-column rule — (AB)ᵢⱼ equals the dot product of row i of A with column j of B — follows directly from this composition logic. Each column of B tells you where a basis vector lands after B's transformation. Multiplying A into that column then applies A's transformation to the result. The inner dimensions must match because B's output must be a valid input for A: if B produces n-dimensional vectors, A must accept n-dimensional inputs, which means A must have n columns.
Non-commutativity is the most striking property. For numbers, ab = ba always. For matrices, AB and BA are often not equal — and often one doesn't even have defined dimensions while the other does. The geometric reason: rotating then reflecting a shape generally gives a different result than reflecting then rotating. Order matters with transformations, so order matters with matrix multiplication.
Associativity, by contrast, holds: (AB)C = A(BC). This corresponds to the fact that when you compose three transformations, it doesn't matter whether you first combine the first two or the last two — you get the same overall transformation either way. This makes long chains of matrix products unambiguous without parentheses. Combined with the dimension-matching requirement, it means you can think of a sequence of matrix multiplications as a pipeline: each matrix passes its output as input to the next, and the final product encodes the full pipeline as a single matrix.