Do we ever put functions as entries of a matrix? If so, are these matrices used in linear algebra or do they have some other special use?
There have been minor not neccessarily conflicts per se, but disagreements on the nature of this question and so I am adding a little statement below to clear this up.
I have noticed that "function" has been interpreted two ways within the answers.
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An actual raw function such as merely writing "f". Such a concept is beyond my current understanding (unless I am being stupid somehow), but it is interesting nonetheless.
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A function call returning a value. This is primarily what I meant in my post.
Either one of these is valid. In fact, I think the broadness of this question dictates the fact that people will interpret it differently. In essence, your mileage will vary, and both are so similar from my standpoint that they are all good answers (or good examples if not standalone answers).
Best Answer
You can define a matrix with elements in any commutative ring, since the only requirement is to be able to perform addition and multiplication with the usual properties.
You even may consider the following $2\times 2$ matrices, with elements that do not belong to the same sets. Such matrices describe the endomorphisms of the direct sum $\;E=U\oplus V$ of two vector spaces $U$ and $V$ $$M=\begin{bmatrix} f_1&f_2\\g_1&g_2\end{bmatrix},\quad\text{where}\quad\begin{array}{|ll} f_1\in \mathcal L(U,U),& f_2\in \mathcal L(U,V),\\ g_1\in \mathcal L(V,U),& g_2\in \mathcal L(V,V). \end{array}$$ You can check one can multiply two such matrices, multiplication of elements being composition of linear maps.