Multiplication beetween matrices with Identity Matrix

Question

I've got this matrix product, where B is non-singular matrix and I is the identity matrix.

$M = BIB^{-1}$

Is it lecit to write $M = BIB^{-1}=IBB^{-1}$ ? If yes, what principle (or theorem) allows me to do it ?

Answer 1

It's simple matrix product. The identity matrix $I$ is the multiplicative unit, meaning the easily verifiable fact that $$IB=B\ \text{ and} \ BI=B$$ for any other $n\times n$ matrix $B$. In particular, $I$ commutes with every matrix, and we indeed have $$BIB^{-1}=IBB^{-1}=I\,. $$