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I am wondering which "bold"-style is the "appropriate" or "official" way for matrix notation. E.g., let's say the matrix "A." \textbf{A} would result in a more italics representation compared\textbf{A} etc.
\boldsymbol{A}\textbf{A}\pmb{A}\boldmath{A}ISO/IEC regulations for technical writing prescribe using bold italic for matrices (and slanted sans serif for tensors). I find these regulations incoherent, but if you work in some fields you are required to follow those regulations. Why incoherent? From a mathematical point of view, a matrix is a tensor.
On the other hand, pure mathematics and theoretical physics mostly ignore those regulations. Doing pure mathematics, I'd not use any special notation for matrices, that are just another mathematical object which a variable is assigned to. For undegraduate textbooks, a special notation is usually employed, in order to help students find their way; upright boldface is mostly used.
Now to the TeXnical part. First of all you should define a personal command:
\newcommand{\matr}[1]{\mathbf{#1}} % undergraduate algebra version
%\newcommand{\matr}[1]{#1} % pure math version
%\newcommand{\matr}[1]{\bm{#1}} % ISO complying version
and use \matr (any other name is possible, of course). This will allow changing just the definition instead of chasing in the document in case you change your mind about the problem.
Note that \boldsymbol (from amsbsy) is obsolete and \bm (package bm) should be used.
Don't use \textbf, because this will inherit font settings from the context, so in a theorem statement you'd get bold italic.
Never ever use \pmb (unless you're in an emergency with some symbol for which there's no other bold version). The syntax \boldmath{A} will do nothing else than issuing two warnings and typeset a normal math italic “A”.
See Consistent macro for bold upright vectors in both latin and greek in case you need also Greek letters for matrices.
About amsbsy and \boldsymbol being obsolete, here's the start of the manual for amsbsy:
\boldsymbol is obsolete" statement? Thank you.
amsbsy; I added a screen shot of the first page.
:)
I honestly don't think there is an official bold typeface for matrices. It is rather an inherited typographical convention from the nineteenth century, when put fashion boldface, arise matrices and vectors and in the absence of typefaces that could easily and properly represent the arrows on the vectors, these and matrices are typeset in bold since then.
I can't assure you that the visual result of the four options you mention is identical, however, the only thing required by this convention is that the name of the matrix will be compound with bold and uppercase letters, regardless of how you get it.
\newcommand*{\matrix}[1]{\mathbf{#1}}, to keep in line with consistent typography; so it is very easy to change one's mind about the particular style of matrix representation.
\textbf and \mathbf (see Why Different Commands for Seemingly Similar Tasks? and this answer or this): \textbf{A_1} will not work, but \mathbf{A_1} will. You might also want to mention the bm package \bm package versus \boldsymbol and Bold math: ....
Applied mathematics, particularly, abounds with long complicated expressions whose basic variables are scalars, (column) vectors, and matrices. To give immediate visual insight into the structure of these expressions, it is useful to maintain a consistent typographical convention to distinguish the three types of variable. This is because the kinds of manipulation that can be performed on each type are different, especially those involved in chains of equalities needed to get a particular result. For example, we may be able to take out of an expression a (square nonsingular) matrix factor, because such a matrix can be inverted; while it would make no sense to "invert" a vector.
A good convention is to use lowercase italic for scalars, lowercase bold italic for vectors, and capital italic for matrices, and similarly for scalar/vector/matrix-valued functions. If it is necessary to use both lowercase and capitals for scalars, then capital bold italic can serve for matrices. The reason for preferring italic for variables is that roman (upright) type signals a term with standard constant meaning, where juxtaposition is a feature of spelling rather than an indicator of multiplication.
