When would you do matrix addition and multiplication to the same matrix? Matrix addition, where we add corresponding elements to eachother, is much more similar to scalar addition, than matrix multiplication, where we add linear combinations of sub-vectors of the matrices, is to scalar multiplication.
It seems to me that matrix multiplication and addition are used for completely different purposes. The typical application of matrix multiplication is to matrices that represent a system of linear equations, but the typical application of matrix addition is to matrices that represent quantities of variables that have some 2 dimensional relation to eachother. An  example of the latter is a matrix who's row represents gender, and who's column represents an age group, and where each entry represents the amount of people in that age group and gender. This example does not represent a system of linear equations, so matrix multiplication seems to be meaningless.
So my question is, is there ever a situation when the matrix addition and multiplication operation are both meaningful on a certain matrix?
 A: Suppose we apply the linear transformations $A$, then $B$, but this application comes with an error (this can occur in signal-processing applications for example, where the transformations may represent the signal passes through a filter or echoes in a room.) If we call the errors $M,N$, we can write the result (for $s$ a signal vector) as $$B(As+M)+N$$
While the errors are unknown they are "small" in relation to $A,B$. This sort of model is used in real applications.
A: Matrices are simply linear functions represented w.r.t a given basis. So if we have matrices A and B then they represent functions which we'll call $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ and $S:\mathbb{R}^k \rightarrow \mathbb{R}^p$ respectively.
Then if $n=k$ and $m=p$, the matrix C = A + B represents the linear map, 
$$
F :\mathbb{R}^n \rightarrow \mathbb{R}^m\\
F(v) := T(v) + S(v)
$$
And if $n=p$, the matrix D = AB represents the linear map,
$$
G :\mathbb{R}^k \rightarrow \mathbb{R}^m\\
G(v):= T(S(v)).
$$ 
Hence multiplication and addition of matrices is so defined as to allow us to 'add' and compose linear transformations entirely using matrices. In this respect the definition of addition and multiplication of matrices is always meaningful.
