A term for "number of columns" of a matrix Is there a single English word to denote the "number of columns" of a matrix?
For example, "dimensionality" of a $2\times 3$ matrix is $2\times 3$. I need a term for $3$ in this example. I can always just say "number of columns", of course, but could I have a single word for it?
 A: I like "width", as suggested by aivanov.  It is difficult to be more specific, without being longer and without using several words.  If several words are ok, then we are back to "number of columns".
An alternative is to reformulate: Instead of saying:

$A$ is a matrix of width 3.

or

The number of columns of $A$ is 3.

why not just say

$A$ has 3 columns.

A: It's not clear why it's important to have only a single-word term.
I'd say that for most purposes don't be afraid of three shortish words: "number of columns"; the very term you began with, which served to perfectly convey the idea you asked about.
It is a mere 5 syllables, it's not a strain to say or write. 
A term consisting of just a few common words is likely to be less of an intellectual effort to grasp on the fly than one longish or unfamiliar one. 
There may well be four or five terms that might serve, but if I was only writing it a few times, I would feel the strong inclination to go back and edit such a term out and replace it with "number of columns". 
If I would need to write about a specific case many times, I'd start with something like "Let $X$ be an $n\times p$ matrix" and then refer to $p$ thereafter; or if I had to refer to the number of columns of many matrices, to have a notation like "denote the number of columns of a matrix $A$ as $c_A$", or something similar as suited the circumstances. 
A: It seems like mostly a specific language (English) problem. In German language this would be much less problematic. The term 'spaltenanzahl' is not a strange word and regularly used. So you may consider introducing 'column-count' or 'column count' (while column-number would be ambiguous), or accept 'number of columns' as not that complex after all (and can often be written simpler with other sentence constructions e.g. 'matrix $M$ has $w$ columns'). 
Some background on the article and examples of sentences would help to look for different terms. One could use width and length or (as Sylvester, origin of the term matrix, did use) breadth and depth. Maybe based on what the matrix actually presents (e.g. system of equations, a polynomial, a vector space, etc.) other terms could be used.
Depending on your article background (or maybe no, independent from the public for your article, maybe only if you do something entirely new) I would advise to not use any newly invented term, and neither use some existing term (which must be an archaic term). 
You have to ask yourself whether the poverty in the English language, not containing a simple synonym for the German term spaltenanzahl, is worth introducing something fancy that may only be confusing.
A: I always recommend avoiding to talk about matrices at all. Most applications that deal with matrices are in principle not interested in matrices at all, but rather in linear mappings between vector spaces. The basis-expanded representation of a mapping $\mathbb{R}^n \to \mathbb{R}^m$ is an $n\times m$ matrix, thus the column-count $n$ is the domain dimension of the mapping.
Likewise, the row count would be the codomain dimension.
These aren't single words, but they're reasonably short yet precise.
A: There is a concept of wide and narrow data, so maybe you could use the term „width“ for the number of columns after you define it in order to avoid the ambiguity. 
A: Let's review your objectives:


*

*You want a short, meaningful term.

*You want it to be memorable and readable, rather than some clunky abstract mathematical or computerese construction like "let $\mathbb{A}\in\operatorname{Mat}(n,p)$" or even "$\mathbb{A}\in\mathbb{R}^{n\times p}.$"

*You want to be able to specify the number of columns explicitly, so you can distinguish (say) 3-column matrices from 2-column matrices or $p$-column matrices from $p+1$-column matrices.

*You want it function as a noun rather than an adjectival phrase; that is, it should read like "$\mathbb A$ is a $p$-column matrix" rather than "the column count of $\mathbb A$ is $p$."

*Evidently, even a short phrase like "$p$-column matrix" is too much!
As others have remarked, you're in the domain of creating your own terminology.  However, "columnity" (which has been proposed) has a grotesque and non-English aspect, albeit being little briefer.
If I were in this position, at the outset of the document I would introduce a term and define it, perhaps like this:

Because we will frequently need to refer to the number of columns in a matrix, let us say that a $p-$matrix is any matrix with exactly $p$ columns.

That seems to meet all the objectives.  It's hard to imagine anything simpler, short of a mathematical notation (which violates objective $(2)$).  It also is consistent with long-standing mathematical terminology, which includes well-known terms like "symmetric matrix," "real matrix," "transition matrix," "rank-$p$ matrix," etc.
A: Personally I would denote the matrix as $$X \in \mathbb{R}^{n \times p}$$ and use $p$ as a reference (assuming your matrix is composed of real values!).
Also note that the notation p >> n is quite widely used to describe the 'short and wide' datasets, e.g. datasets where the number of rows (observations) is significantly lower than number of columns (features). There is an area of Statistics known has 'High-dimensional statistics' that deals with these kinds of problems. 
A: I propose you do as Tukey would have done and invent a word. It is of course OK to define new terminology as long as we are explicit about it. As you say it might not gain immediate traction, but it would still work within the scope of your paper. My personal suggestion is

columnity [n.] of A: the extent to which A is columnar

An edit: Thinking a bit about my light-hearted suggestion here I felt I should also add that width is my preference. It's short and sweet and follows the excellent tip in Orwell's Politics and the English language to never use a long word where a short one will do.
A: go for width \ height as mentioned
its pretty clear what you mean and even a child knows what those words mean
(providing the array is always represented from the same viewpoint)
Of course if you go into arrays of more than 3 dimensions (height, width, depth) it gets a little tricky and is probably better to use matrix notation
A: There does exist and English word columnarity which means the property of being columnar. So saying a matrix of columnarity 3 would seem quite natural.
A: I am guessing that you use your matrix for representing data. Usually, columns represent the different features and rows are different data points, as the data store in the figure below (ref).

This then extends to the the dimension of useful matrices, e. g. for mixing these features into a new dimension.
In that context, the number of columns is the number of dimensions of your feature space, that is, simply dimensionality.
