Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Digital systems, linear algebra, introduction to formal proof / 4 / a / i

By matrix multiplication, $D_{ij}={\sum }_{k=1}^N{B}_{ik}{C}_{kj}$. A potentially nonzero term requires $k=i$ because $B$ is a diagonal matrix, and $k=j$ because $C$ is a diagonal matrix. For $i\ne j$ these requirements cannot both hold. Thus every summand is zero and $D_{ij}=0$ whenever $i\ne j$.