Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes

Let $F$ be a field, let $m,n,p\in\mathbb N$, and let $A=(a_{ij})\in M_{m\times n}(F)$ and $B=(b_{jk})\in M_{n\times p}(F)$. Their matrix product is the matrix $AB\in M_{m\times p}(F)$ whose entries are

$(AB)_{ik}:=\sum_{j\in n}a_{ij}b_{jk}.$

The sum is the finite sum in the additive commutative monoid of $F$. If $n=0$, it is the empty sum, so $AB$ is the zero matrix of shape $m$ by $p$.

The identity matrix $I_n\in M_n(F)$ has entry $1$ when $i=j$ and entry $0$ when $i\ne j$. In particular, $I_0$ is the unique empty matrix in $M_0(F)$.