Question

Two $n\times n$ square matrices $A$ and $B$ are said to be similar if there exists a non-singular matrix $P$ such that $P^{-1}AP=B$

If $A$ and $B$ are two non-singular matrices, then

• A. $A$ is similar to $B$
• B. $AB$ is similar to $BA$
• C. $AB$ is similar to $A^{-1}B$
• D. none of these

Answer 1

Answer: (B)

$AB$ is similar to $BA$

$AB=\left(B^{-1}B\right)AB=B^{-1}\left(BA\right)B$

$\therefore$ $AB$ is similar to $BA$.

Hence, option B.