Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 4 / b

The product $AB$ is $m\times p$, so multiplying it by the $p\times q$ matrix $C$ gives an $m\times q$ matrix. The product $BC$ is $n\times q$, so multiplying it on the left by the $m\times n$ matrix $A$ also gives an $m\times q$ matrix. Both multiplications have matching inner dimensions.

For the specified row and column, expand the two matrix products and reorder only finite sums:The middle equality uses scalar associativity and distributivity, and the interchange of two finite sums. Together with their equal dimensions, equality of every entry proves $(AB)C=A(BC)$.