
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 $\displaystyle A$ be an $\displaystyle m\times n$ matrix and $\displaystyle B$ be an $\displaystyle r\times s$ matrix where:
$\displaystyle A=\begin{pmatrix}a_{1,1} & \cdots & a_{1,n} \\ \vdots & \ddots & \vdots \\ a_{m,1} & \cdots & a_{m,n}\end{pmatrix}$
$\displaystyle B = \begin{pmatrix}b_{1,1} & \cdots & b_{1,s} \\ \vdots & \ddots & \vdots \\ b_{r,1} & \cdots & b_{r,s}\end{pmatrix}$
If, as you suggest, scalar powers work like this:
$\displaystyle A^k = \begin{pmatrix}a_{1,1}^k & \cdots & a_{1,n}^k \\ \vdots & \ddots & \vdots \\ a_{m,1}^k & \cdots & a_{m,n}^k\end{pmatrix}$
then you can define matrix powers using matroids as this:
$\displaystyle A^B = \begin{pmatrix}A^{b_{1,1}} & \cdots & A^{b_{1,s}} \\ \vdots & \ddots & \vdots \\ A^{b_{r,1}} & \cdots & A^{b_{r,s}}\end{pmatrix}$
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.