Definition of a Matrix:
A matrix is a rectangular arrangement (array) of objects enclosed in brackets (square or round).
Let me be specific with my question, I have no trouble understanding the part that it's a "rectangular" array of objects. But my question is, is it necessary for a matrix have brackets?
For instance, can I call this thing below,
$$ \begin{array}{cc} a&b\\ c&d\\ e&f\\ \end{array} $$
or maybe this simply (table/grid) a matrix in mathematics?
On
The questions being asked here belie a misunderstanding about what mathematics is.[1] The question asks if the definition of a matrix is correct. This question doesn't really have a good answer: in mathematics, a definition creates or describes a new kind of object. A definition can say whatever we want it to say. For example:
Definition 1: A flübwizzle is a wizzle which has the additional property that it can be oriented and has length $3$.
Presumably, I've already told you what a wizzle is, and what it means to be oriented, and what length means, but none of that really matters here. This definition is "correct" in that it obeys the laws of English grammar in mathematical writing, and is a syntactically correct definition.
The better question is: "Is this definition useful?"
In higher mathematics, matrices can be thought of in a number of different ways. In a very abstract setting, a matrix is an element of an algebra, where an algebra consists of a field (a set of objects where addition and multiplication "make sense") and another set of "algebra objects" or matrices. An algebra has some additional structure: there are two kinds of multiplication (a matrix be multiplied by a field element—this is scalar multiplication; two matrices may be multiplied—this is matrix multiplication), and there is a matrix addition.
Alternatively, a matrix can be seen as a way of representing a linear transformation between vector spaces, where a vector space (like an algebra) is a field and a collection of objects (call vectors) where there is are scalar multiplication and vector addition operations, and a linear transformation is a function which respects these two operations.
If one wanted to have a really, really useful definition of a matrix, then one should take one of the ideas above, and write it up more rigorously. Typically, coming to terms with these ideas will be the topic of one or more chapters in a textbook on abstract algebra, so there is quite a bit of work to be done.
However, there are reasons to study matrices without necessarily needing all of that heavy theory. For example, in American schools, matrices are often introduced at the end of high school as a way of organizing data for solving linear systems of equations (this is, by the way, related to the notion of a matrix as a linear transformation between vector spaces). It is often best not to throw too much abstract theory at high school students, so it is reasonable to define a matrix as has been done above, i.e.
Definition 2: A matrix is a rectangular arrangement (array) of objects enclosed in brackets (square or round).
While I agree with Hagen von Eitzen's that this is really more of a notation than a definition, it can be a workable definition. Rather than starting from the abstract end of things, it makes sense to define a matrix by its notation, then introduce addition and multiplication. So this definition is fine.
Note, however, that a mathematical object is exactly what it is defined to be. So if this is your definition of a matrix, then a matrix must have brackets of some kind. If it didn't have brackets it wouldn't be a matrix.
Of course, we could also write
Definition 3: A matrix is a rectangular arrangement (array) of objects.
This definition does not require brackets, and (as with my definition of a flübwizzle) is a perfectly correct mathematical definition (it is grammatical and everything). Similarly, I could bake any other form of notation into my definition (it must be a grid; there must be lines around all of the elements; whatever).
Definitions 2 and 3 are very similar, so the next question is: "Which is more useful?"
From some point of view, it doesn't really matter. However, it seems likely that matrices are being defined so that we may eventually do things with them, such as multiply them by scalars, add them together, or multiply them. As Antonios-Alexandros Robotis points out in their answer, a bracket-less notation could cause confusion. For a more clear example, suppose that $$ A = \begin{matrix} a & b \\ c & d \end{matrix}, \qquad\text{and}\qquad v = \begin{matrix} x \\ y \end{matrix}, $$ where $A$ is a matrix and $v$ is a vector (thought of as a column vector, or a $2\times 1$ matrix). Is $$ \begin{matrix} a & b \\ c & d \end{matrix} \begin{matrix} x \\ y \end{matrix} $$ the product $Av$, or is it a $2\times 3$ matrix? Without brackets, it is very hard to tell. Therefore, while Definition 3 is not a wrong definition, I would suggest that it is a bad definition, because it doesn't let us do the things we want to do.
In mathematics, we write definitions so that they allow us to do the kinds of things that we want to do as simply and unambiguously as possible. When you encounter a new definition which seems to have unnecessary requirements (such as brackets for a matrix), the question is not whether the definition is right or wrong, but rather if the definition is useful. If you don't understand why an author included something in their definition, then it is good to question it—take some time to work out some examples, and see if you can work with a less restrictive definition.
In this case, the author is attempting to give a definition of a matrix which is (1) accessible to a relatively elementary audience and (2) which allows for the kinds of manipulations which will likely be introduced later. There are good reasons to use matrix notation with clear delimiters, since we eventually want to multiply matrices together, and delimiters are likely going to be a notational necessity, in that setting.
[1] The is, to be clear, not the fault of the asker. I have a whole rant prepared about how the American educational system destroys mathematics by teaching formalism and notation above reasoning—I imagine that other educational systems suffer from a similar flaw. However, that is off-topic here, so I'll refrain.
While you could perhaps assume this notation without changing any mathematical content, the notation is inherently confusing. For instance, if we take $$ A=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{bmatrix}$$ and write $A^2$ in your notation, it looks like: $$ A^2= \begin{matrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{matrix} \begin{matrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{matrix} $$ which is unsightly and confusing. So it is much better to use brackets.