I want to say that squaring is a form of scaling so that it should be true; however I can't make sense out of it and clearly see why.
The problem is that while it's a form of scaling, if we think about a matrix as a list of coefficients of variables of a system of equations, then the coefficients are being multiplied among all the variables.
For example, with the simple scaling by a single value of a matrix, a system of equations would retain the same exact solutions it had before scaling.
However, when squaring the matrix (I tried with a specific example), the solutions differ in the squared version of the system.
Best Answer
Yes. Moreover $(A^2)^{-1}=(A^{-1})^2.$ Note that
$$(A^{-1})^2A^2=A^{-1}A^{-1}AA=A^{-1}IA=A^{-1}A=I.$$