Inverse matrices properties.

Question

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ in the equation $A= PBP^{-1}$, where $P$ and $B$ are matrices and $P^{-1}$ is the inverse of $P$?

Answer 1

Algebras satisfying the identity $(ab)c=(ac)b$ are a well-known special class, i.e., they have commuting right multiplications: $R_bR_c=R_cR_b$, or $[R_b,R_c]=0$. Matrix algebras do not have commuting right multiplications in general.

The equation $A=PBP^{-1}$ can be transformed easily. We obtain $AP=PB$ and then $B=P^{-1}AP$ by multiplying with $P$ and $P^{-1}$. Also $A^n=PB^nP^{-1}$ for all $n\ge 1$.