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 similar and $B$ and $C$ are similar, then

• A. $AB$ and $BC$ are similar
• B. $A$ and $C$ are similar
• C. $A+C$ and $B$ are similar
• D. none of these

Answer 1

The correct option is B $A$ and $C$ are similar

$A=P^{-1}BP,B=Q^{-1}CQ$,

$\Rightarrow A=P^{-1}\left(Q^{-1}CQ\right)P=(QP{)}^{-1}CQP$

Thus, $A$ is similar to $C$

Hence, option B.