## anonymous 5 years ago Let V be a finite dimensional vector space over the field of complex numbers (C), and let T be an invertible linear operator on V. Prove that if c doesn't equal 0 is an eiganvalue of T, then 1/c is an eiganvalue of T^-1

Suppose $$x$$ is the eigen vector of $$T$$ corresponds to the eigne value $$c$$. Then $$T(x)=cx$$ Now $$x=T^{-1}(T(x))=T^{-1}(cx)=cT^{-1}(x)$$ Therefore $$T^{-1}(x)=\frac{1}{c} x$$.