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

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

2020-11-09
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 PSKAB=AB(AA^-1) =A(BA)A^-1ZSK
This shows that AB and BA are similar.

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