I am an engineer who is working with some linear equation problems.
In my application, I found out that in having the symbolic form of such matrix inverse actually speed things up (for example ${A^{ - 1}} = \frac{1}{{\det \left( A \right)}}\left[ {\begin{array}{*{20}{c}}d&{ - b}\\{ - c}&a\end{array}} \right]$). Therefore, it is natural for me to raise the question about the scaling aspect of this approach.
Note that, modern computer algebra system supports symbolic evaluation of matrix inverse and I have very powerful computer.
What would be the complexity of conducting a symbolic evaluation of matrix inverse?
I have been looking for everywhere on the internet and cannot find the answer to this problem. Even worst, I feel that complexity of symbolic computation does not receive adequate attention within the literature.
Thank you for your enthusiasm!