 Associativity of matrix multiplication

For dimension-compatible matrices, $(AB)C=A(BC)$. In coordinates both sides have $(i,j)$ entry ${\sum }_{k,l}{A}_{ik}{B}_{kl}{C}_{lj}$; distributing and regrouping the finite sums proves equality. This permits accumulating a product left to right even when a recursive algorithm groups the same transformations from the right. It does not permit changing factor order, because matrix multiplication is generally not commutative.