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.