Question

If T and S are similar then prove that T^(2) and S^(2) are also similar .Further if T and S are invertible then prove that T^(-1) and S^(-1) are also similar.

Answer 1

Let $T$ and $S$ be similar. Then $T=A^{-1}SA$ for some invertible $A.$ It follows that $T^2=TT=(A^{-1}SA)(A^{-1}SA)=A^{-1}S(AA^{-1})SA=A^{-1}SSA=A^{-1}S^2A,$ and hence $T^2$ and $S^2$ are also similar. If $T$ and $S$ are invertible, then $T^{-1}=(A^{-1}SA)^{-1}=A^{-1}S^{-1}(A^{-1})^{-1}=A^{-1}S^{-1}A,$ and we conclude that $T^{-1}$ and $S^{-1}$ are also similar.