# Formula for the $(i, j)$-th entry of the product of 3 matrices?

Is there a formula for the product of 3 matrices? That is, if $$A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times n},$$ and $$C \in \mathbb{R}^{n \times p}$$, and I want the $$(i, j)$$ entry of the product $$D = ABC$$, how can I write $$D_{i,j}$$? I know $$(AB)_{i,j} = \sum_{k=1}^na_{ik}b_{kj}$$, but I'm not sure if this can be generalized to more than 2 matrices. Thanks for any help.

You can done it for a arbritary number of matrices, the case of tree matrices goes $$(AB)_{i,j} = \sum_{k} A_{i,k} B_{k,j}$$ With some renaming we reach $$(XY)_{a,b} = \sum_{c} X_{a,c} Y_{c,b}$$ So we replace $$X$$ with $$AB$$ $$((AB)Y)_{a,b} = \sum_{c} (AB)_{a,c} Y_{c,b}$$ $$((AB)Y)_{a,b} = \sum_{c} (\sum_{k} A_{a,k} B_{k,c}) Y_{c,b}$$ After some basic manipulations we get $$(ABY)_{a,b} = \sum_{c} \sum_{k} A_{a,k} B_{k,c} Y_{c,b}$$ Finally replace $$Y$$ with $$C$$ $$(ABC)_{a,b} = \sum_{c} \sum_{k} A_{a,k} B_{k,c} C_{c,b}$$ So we had derivted our triple matrix formula. (And like this we can generalize for n matrices)
It would be similar to the 2 matrices case but it involves 2 nested sums. I am not sure if this is very efficient in practice: $$d_{ij} = \sum_u\sum_va_{iu}b_{uv}c_{vj}$$