 Solution

ID: 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/ii/solution

For the specified row and column, expand the two matrix products and reorder only finite sums:

((A B) C)_{ij} = k = 1 \sum p (ℓ = 1 \sum n A_{i ℓ} B_{ℓ k}) C_{kj} = ℓ = 1 \sum n A_{i ℓ} (k = 1 \sum p B_{ℓ k} C_{kj}) = (A (BC))_{ij} .

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 $(A B) C = A (BC)$ .

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