Associativity of matrix multiplication

Statement

Let $R$ be a ring and let $m, n, p, q \ge 0$ be integers. Let $A$ be an $m \times n$ matrix, $B$ an $n \times p$ matrix and $C$ a $p \times q$ matrix over $R$. Then $(AB)C = A(BC)$, where the products are matrix products.

Remarks

Both sides have entry $(i, l)$ equal to $\sum_{j=1}^{n} \sum_{k=1}^{p} a_{ij} b_{jk} c_{kl}$. Because of this property, the product $ABC$ needs no parentheses. The property also follows from the associativity of the composition of linear maps.

Explanations

L1 human

When we multiply three matrices, we may multiply the first two first, or the last two first. The result is the same in both cases. This is like $(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4)$ for numbers.