Use the definition of inverse.
The inverse of A^2, call this B, has the property B * A^2 = I
Now let B = (A^-1)^2
This was a question on one of my prelims for linear algebra that I'm reviewing for the final:
If A has an inverse, then so does A squared and the inverse of A squared = A inverse, squared:
(A^(2))-1 = (A^(-1))2
I started to prove that A squared has an inverse but didn't make much headway. What should I have done?
Always, if you already have a guess for the inverse, just multiply the inverse guess by the original matrix, if the result is the identity matrix, then you have shown conclusively it is the inverse. A more low level proof would be something like this:
We know that A^{-1} exists.
This means that A^{-1} * A^{-1} = (A^{-1})^2 exists.
Now we have
(A^2) * (A^{-1})^2 = A * A * A^{-1} * A^{-1} = A * I * A^{-1} = A * A^{-1} = I
Thus, (A^{-1})^2 satisfies the definition of the inverse of A^2 or (A^2)^{-1}
so (A^2)^{-1} = (A^{-1})^2
(Every thing posted before is correct, but I just thought it also good to provide a proof using basic definitions)