Is there a program that can take matrices of variables and compute the inverse?

Question

I have a control theory / optimal control problem that is formulated like this:

$A(x)\dot x = f(x,u)$

Where A(x) is a matrix containing state variables. And I want to isolate $\dot x$ by doing this:

$\dot x = A(x)^{-1}f(x,u)$

So that I have state rates equal to states and inputs. The issue is inverting A(x), it's a 12x12 matrix with probably 100 or more instances of states. Is there a program that will automatically output the inverse matrix or, even better, the entirety of:

$A(x)^{-1}f(x,u)$

I currently know f(x,u) and A(x). Could someone give me the name of a free program or the type of program I'm looking for?

Answer 1

The matter is that if A has so many variables, even with very simple functions of them, any CAS (even the most powerful one) either will refuse to work or will output such a tremendously complicated output that it will be useless.

That unless the matrix is highly symmetrical and simple, that it can be simply LU decomposed, or has simple blocks and /or eigen-values, eigen-vectors etc.