inverse of abstract matrix

Question

If you assume the matrix $A$ is invertible, then $A^{-1} \cdot A = I$. Is there an assumption for invertibility in Mathematica 9? How can one make the following evaluate to the identity matrix $I_3$?

$Assumptions = {Element[A, Matrices[{3, 3}]]}
Inverse[A].A

Answer 1

This gives what you want! Adding the axiom of matrix power $A^0=I$ to Simplify gives us

Simplify[
Assuming[Element[A, Matrices[{n, n}]] && Det[A] != 0,TensorExpand[Inverse[A].A]], 
ForAll[{A}, MatrixPower[A, 0] == IdentityMatrix[n]]]

IdentityMatrix[n]

Use n=3 to validate your case.