@ if A^-1(inverted) matrix exists and B matrix and S matrix prove these AB=BA and S^2 =B prove that (A^-1SA)^2=B

Question

@ if A^-1(inverted)  matrix  exists and B matrix and S matrix prove these AB=BA and S^2 =B prove that (A^-1SA)^2=B

jamesj · Answer

1. A is invertible
2. AB = BA
3. S^2 = B

Show that $ (A^{-1}SA)^2 = B $.

So just start calculating:
$$ (A^{-1}SA)^2 = A^{-1}SAA^{-1}SA = ... $$

zarkon · Answer

nice translation of the question ... :)

jamesj · Answer

Thanks. ;-) I guess I shouldn't be surprised, but it is interesting how often clarifying the question to "well posed" is necessary on here.

anonymous · Answer

so in the end it is ...=I*B am I right ?

jamesj · Answer

and IB = B, by definition of I