# Solve for element $\gamma(i,j)$ in matrice multiplication.

Let $A$ and $B$ be $100 \times 100$ matrices. Elements in these matrices are $\alpha_{i,j} = i + j$, $\beta_{i,j} = i - j$. In a multiplication matrix $C = AB$, what is the element $\gamma(i,j)$, as function of the indexes $i$, $j$?

I looked into the definition of matrice multiplication:

$(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$

But I don't understand how to use this to solve the problem.

The answer is $γ_{ij} = 385 + 55(i − j) − 10ij$, but I'm more interested in how to get there.

$(AB)_{ij}=\sum_{k=1}^{100} \alpha_{ik}\beta_{kj}=\sum_{k=1}^{100} (i+k)(k-j)$
For example, the entry of row 5, column 3 is: $\sum_{k=1}^{100} (5+k)(k-3)$