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On the one hand, I read on Wikipedia that

[A] matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

However, googling "3D matrix" produces a huge number of hits, including many from this website.

I also find some people complaining about alleged misuse of terminology, e.g. here

Just to gratify a pet peeve, there is no such thing as a "3D matrix". Matrices are 2D by definition.

What's the most appropriate terminology here, say for a high school or undergraduate mathematics teaching context?

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    $\begingroup$ In as much as a matrix represents a (linear) transformation, it represents a function, having a domain and a codomain. The "2-ness" of the domain-codomain pair is why the matrix is 2-dimensional. $\endgroup$
    – Simon
    May 31, 2016 at 8:51
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    $\begingroup$ The "original" definition of Matrix is : "a rectangular array of numbers, symbols, or expressions, arranged in rows and columns." But it can be easily generalized : "Another extension are tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers." $\endgroup$
    – Mauro ALLEGRANZA
    May 31, 2016 at 8:56
  • $\begingroup$ @Simon I disagree: A matrix by itself is completely seperate from the concept of a linear transformation. Matrices can be used to describe them, though. Matrices can be used in different contexts as well e.g. storing an image. $\endgroup$
    – Stefan Hante
    May 31, 2016 at 8:59
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    $\begingroup$ @Wauzl You're right in general. On the other hand, if $K$ is a field and $M_n(K)$ the set of all $(n\times n)$-matrices over $K$ then each matrix $M\in M_n(K)$ defines a linear transformation on the $n$-dimensional vector space over $K$. $\endgroup$
    – Dr_Be
    May 31, 2016 at 9:10
  • $\begingroup$ @Wauzl, by writing "In as much as" I meant "to the extent that a matrix DOES represent a linear transformation". That is, I meant to allow that matrices can be used in many other ways, too. There is an nice video here "The Rise and Fall of Matrices" by Prof. Walter Ledermann for the London Mathematical Society which explores the history of the relationship between matrices, determinants and linear operators: youtu.be/57Guhkvglm0 $\endgroup$
    – Simon
    May 31, 2016 at 9:36

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The term "$3$-D array" is acceptable in most fields; this term is more common in computer science. See: array.

Another term you can use for these from the math/physics side is "tensor". That being said, tensors are not exactly just boxes of numbers, even if it is possible to think of them as such.

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