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 matrices such that $det\left(AB\right)=0$, then

• A. $det\left(A\right)=0anddet\left(B\right)=0$
• B. $det\left(A\right)=0ordet\left(B\right)=0$
• C. $A=0andB=0$
• D. $A=0orB=0$

Answer 1

Answer: (A)

$det\left(A\right)=0anddet\left(B\right)=0$

As $A$ and $B$ are similar matrices there exists a non-singular matrix $P$ such that $A=P^{-1}BP$

$\Rightarrow det\left(A\right)=det\left(P^{-1}BP\right)$

$=det\left(P^{-1}\right)det\left(B\right)det\left(P\right)$

$=\frac{1}{det\left(P\right)}det\left(B\right)\left(detP\right)$

$=detB$

$det\left(AB\right)=0$ $\Rightarrow detA\cdot detB=0$

$\therefore$ $det\left(A\right)=0$ and $det\left(B\right)=0$.

Hence, option A.