Matrix multiplication ambiguity From this source here, it says that matrix multiplication is given by this:
$AB = \begin{bmatrix}
a_{1,1}b_{1,1}+a_{1,2}b_{2,1}+...+a_{1,n}b_{p, 1} & ...\\ 
\vdots  & a_{m,1}b_{1,p}+a_{m,2}b_{2,p}+...+a_{m,n}b_{n,p}
\end{bmatrix}$
where the dimension of Matrix $A$ is $m\times n$ and dimension of Matrix $B$ is $n\times p$. 
If we look at the top left element of $AB$, the final number in the sum ($a_{1,n}b_{p, 1}$).
It seems to me that this is very ambiguous, as it would only work if $n=p$. Otherwise we must stop at the point n or p, given whichever is smallest. 
Can anyone clarify this with me, and if I am correct could someone maybe link me to a less ambiguous definition of matrix multiplication.
Thanks for any help in advance.

Indeed, the final term in the top left position should be $a_{1,n},b_{n,1}$, with an $n$ and not a $p$
