
Matrix^Matrix
You can multiply, add, and subtract matrices and I know if you have a scalar exponent you take every term in the matrix to that power but can you have a matrix exponent and get a defined answer?
[−2 3 4 ] ^[1 −2 2 ]
[1 4 −2] [−1 0 0]
as an example using 2 2x3 matrices.

Re: Matrix^Matrix
Your definition of "scalar exponents" of matrices is incorrect, for example:
$\begin{bmatrix}1&2\\3&4 \end{bmatrix}^2 = \begin{bmatrix}7&10\\15&22\end{bmatrix}$ NOT $\begin{bmatrix}1&4\\9&16\end{bmatrix}$.
It is technically possible to "define" for 2 matrices:
$M^N = \exp(N \log M)$
using "power series" of matrices, but convergence is an issue, and we lose a lot of the properties of exponents we're "used to" unless $M$ and $N$ commute. For this to have any hope of succeeding, we also need to use SQUARE matrices (so we can take powers), and we get "many possible $\log$ functions" (one for each eigenvalue of $M$). This gets a little tricky and complicated, and I would suggest postponing your question for a few years.

Re: Matrix^Matrix
but I have learned some linear algebra like operating with matrices(except for powers) and vectors(including cross product)

Re: Matrix^Matrix
Let be an matrix and be an matrix where:
If, as you suggest, scalar powers work like this:
then you can define matrix powers using matroids as this:
But, again, this is not standard. In general, scalar powers don't work that way, so it seems silly to consider defining matrix powers in this way.