Does matrix addition give you a matrix or a number? I am very confused by something our lecturer said today:
We were given two matrices:
$B=\begin{pmatrix}2 & 3\\ 2 &0 \\ 0&3\end{pmatrix}$
C=$\begin{pmatrix}6 &3&4\\6&6&0\end{pmatrix}$
$\space$
And we were supposed to find: $B_{ij}+C_{ji}$
To me, this seemed like another way of writing $B+C^T \implies \begin{pmatrix}2 & 3\\ 2 &0 \\ 0&3\end{pmatrix}+ \begin{pmatrix}6 & 6\\ 3 &6 \\ 4&0\end{pmatrix}=\begin{pmatrix}8 & 9\\ 5 &6 \\ 4&3\end{pmatrix}$
Our lecturer said that when adding the two matrices I don't get a matrix, I actually get an entry. She also added that the correct way of writing the answer would be: $$\begin{pmatrix}8 & 9\\ 5 &6 \\ 4&3\end{pmatrix}_{ij}$$
How is that not a matrix? Can anyone explain what she means by this?

Usually when one writes $A_{ij}$, one means the entry of $A$ at position $(i,j)$. For example 
$$\begin{pmatrix}6 &3&4\\6&6&0\end{pmatrix}_{2,2} = 6$$
