Question

# Prove that: If A or B is nonsingular, then AB is similar to BA

Matrix transformations
Prove that: If A or B is nonsingular, then AB is similar to BA

We know two matrix M and N are said to be similar if thee exist an invertible matrix P such that $$\displaystyle{P}^{{-{{1}}}}{M}{P}={N}$$
Suppose that A is not singular. Therefore $$\displaystyle{A}^{{-{{1}}}}$$ exist. Now $$AB=AB(AA^{-1}) =A(BA)A^{-1}$$