I would love some direction and any information for some further reading about a cool thought I just had. If a square matrix has functions as the elements, would it not make the matrix depend on $x$ (or whatever variables) and hence have a changing determinant? What would be even cooler is for a matrix to sometimes have a determinant that is equal to $0$. Now that I think about it, this could be a rotation matrix, right? $A_{11} = \cos x$ and $A_{12} = -\sin x$ and $A_{21} = \sin x$ and $A_{22}= \cos x$ so if $x=45^\circ$, the determinant will be $0$ I think. This would mean it has no inverse for that value? I think I made a mistake because it is easy to visualize how to reverse a $45^\circ$ rotation.

In any case, please do comment anything that you know about this and where to find out more about it.

Edit: Yes the rotation matrix was a bad example because the determinant is always $1$. I could have used a better example with a determinant of $\cos x - \pi$ or $2x-6$