Suppose that A and B are similar matrices. Prove that A3 and B3 are similar matrices. Generalize

By Definition SIM we know that there is a nonsingular matrix S so that A = S⁻¹BS. Then

A³ = (S⁻¹BS)³

= (S⁻¹BS)(S⁻¹BS)(S⁻¹BS)

= S⁻¹B(SS⁻¹)B(SS⁻¹)BS                                    Theorem MMA

= S⁻¹B(I₃)B(I₃)BS                                             Definition MI= S⁻¹BBBS                                                      Theorem MMIM

= S⁻¹B³S

This equation says that A³ is similar to B³ (via the matrix S).

 More generally, if A is similar to B, and m is a non-negative integer, then A^m is similar to B^m. This can be proved using induction (Proof Technique I).