Question: solving 4 variables by using a system consisting of 4 (independent...) equations

Dear all,

I'm currently facing the following problem. I am trying to solve an expression for 4 variables: Fe11, Fe12, Fe21, Fe22. Actually these variables come from a 'deformation gradient tensor' Fe that is used in continuum mechanics.  This tensor actually appears in two equations in tensor-form. Since these tensorial equations contains 'inverse' and 'transpose' operators, the only thing I could think of to solve for a particular tensor (Fe), is to do this component-wise. Doing so, results in the following chaotic equations written in the following file:

solvingFor4vars.mw

I hope you like it to look at this problem. In that case, I would like to further explain the problem:

I'm trying to solve a system of equations for the 4 variables in the last line. I think these equations are independent, but if you have other thoughts, would you be so kind to let me know. When I want to solve this system, it takes forever, and all memory is captured gradually. I even tried this at a computer with 24GB of internal memory, and even that was all taken after a while. The thing is that I really dont know what to do.... there are no messages given by maple. Should I change the solution method? Do I have to solve this numerically? Is it solvable?

If you have any thoughts, please let me know. Appreciate it! Thanks a lot!

Bas