How do I know which indices to raise or lower in a rank-$2$ tensor? The document I am reading about tensors claims the following:

In practice it will turn out to be very useful to also introduce this convention [raised and lowered indices] for matrices. Without further argumentation (this will be given later) we note that the matrix $A$ can be written as: $$A:\quad{A^\mu}_{\nu}.\tag{2.19}$$ The transposed version of this matrix is $$A^T:\quad{A_\nu}^\mu.\tag{2.20}$$

However, I cannot find this “further argumentation.” When I have a rank-$2$ tensor, how do I know which indices to raise, if any, and which indices to lower, if any? Does the raising of one index and the lowering of the other indicate that the object is a matrix specifically?
As I understand it, one raised and one lowered index is supposed to evoke that the object has mixed variant and contravariant transformational properties. I can not be for sure, however, because the paper mentions every mixture of index positions.
Edit: If you could explain how, when looking at a $3\times3$ rank-$2$ tensor / array, one knows where to locate the indices, that would also be very helpful. For example, when dealing with a contravariant/column vector, I immediately know the index is raised (at least before manipulation).
 A: Hmm this is a good question! I mean to make things a bit easier...I have always heard that the best way to denote the different placement of the indices with dots...so instead of just having $A^\mu_{\phantom{...}\nu}$ or $A^{\phantom{...}\nu}_{\mu}$ its always goot to put a dot above the index which has been altered. So for example if you have the matrix $A^\mu_{\phantom{..}\nu}$ and lower the first index, then the result is $A^{\bullet}_{\mu\nu}$ and if you then raised the second, the result is $A^{\bullet\phantom{.}\nu}_{\mu\bullet}$...thats just how I learned though! So up to you whatever feels good. The only issue that could happen if you don't is that for some tensors, the position of the indices is VERY important so for example things could get confusing if you dont put dots with tensors such as $A^{\mu\phantom{..}\nu\phantom{..}\sigma}_{\phantom{..}\lambda\phantom{..}\zeta}$ (just as an example). So I will be doing this
Now for example I suppose the best way to prove this for tensors is just to show you the linear algebra 'sense' of it all. We know that a transpose will essentially 'reflect' all the components of the matrix. This makes it easy for a Rank 2-tensor of a single order ($A^{\mu\nu}$ or $A_{\mu\nu}$)...for example:
$$(A^{\mu\nu})^T=A^{\nu\mu}$$ but when it is mixed things are trickier. The only way I can think of explaining it is that you know that the metric tensor and its inverse raises and lowers indices and you also know that:
$$(AB)^T=B^TA^T$$
The cool thing about dealing with matrix components is that you dont need to worry about the placement of the components (in the sense of which matrix is multiplied first) so consider $A^\mu_{\phantom{..}\nu}=g^{\mu\sigma}A^\bullet_{\sigma\nu}$ and then transopose each individually (since the metric tensor is symmetric it is its own transpose)...youll find that you'll get the book's result! Keep at it though! Thats the document I used myself as my introduction to tensor calculus
A: So here I am reading that same pdf/book with a very similar question. 
What I think the answer is (main reason I’m posting this is so that someone who knows better can correct if wrong) as follows:
As we can infer from the pdf ${(A^T)_\mu}^{\nu} = {A^\nu}_\mu$. (Page 13, eqn 2.22)
So switching the left/right order of indeces is equivalent to transposition. Therefore if we consider the original array/matrix/rank 2 mixed upper lower index tensor A, the index further to the left represents the “row” and the one to the right the “column”.
This convention essentially renders redundant the traditional transposition notation as ${(A^T)^\mu}_\nu = {A_\nu}^\mu$ (replace $A^T$ for $A$ in eqn above)
When doing a summation it always has to be with one upper and one lower index, e.g. ${A^\mu}_\nu x_\mu$. I do find it more intuitive to have the indeces being summed over adjacent to each other and therefore sometimes write ${({A^T})_\nu}^\mu x_\mu$.
This seems easier to visualise, but then again I just discovered tensors this week. Other than this visualisation it seems there maybe rarely really is a need to use the ${A_\mu}^\nu$ way of writing and one can mostly just stick to ${A^\nu}_\mu$
But I have to admit I am very confused, because equation 3.3 on page 15 suggests that ${A^\mu}_\alpha = {(A^T)^\mu}_\alpha$. I’ve just assumed this is a typo
