Prove that $(I-ix)^{-1}(I+ix)(I-ix)(I+ix)^{-1} = I$

Question

I have a question

Prove $$(I-ix)^{-1}(I+ix)(I-ix)(I+ix)^{-1} = I$$

with $x$ being a $n \times n$ matrix.

Answer 1

Note that $$(I+ix)(I-ix) = I + x^2 = (I-ix)(I+ix)$$ Now you are done.