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

Question
Matrix transformations
asked 2020-11-08
Prove that: If A or B is nonsingular, then AB is similar to BA

Answers (1)

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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