So I know that transformations follow a similar idea of checking for addition and scalar multiplication but I have not been able to successfully apply that logic to this problem. I do not understand how to check these choices for linearity. Can someone please help me understand how I should approach these problems so I can apply that logic to other similar questions?
Well as an example, for option A....
$T(A)=A^T$ is linear. You can show this by talking about components.
$T(c a_{ij})=ca_{ji}$ or $cT( a_{ij})=ca_{ji}$
$T(a_{ij} + b_{ij})=a_{ji}+b_{ji}$ or $T(a_{ij} )+ T(b_{ij})=a_{ji}+b_{ji}$
For option C...
Any linear transformation can be represented by a matrix (and vise versa). So let $A^6=B$
Then, $T(A)=A^7=A^6A=BA$. So it's linear too.
Try applying these techniques to the other ones.