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 $det\left(A\right)=1$, then

• A. $det\left(B\right)=1$
• B. $det\left(A\right)+det\left(B\right)=0$
• C. $det\left(B\right)=-1$
• D. none of these

Answer 1

The correct option is A $det\left(B\right)=1$

As $A$ and $B$ are similar matrices there exists a non-singular matrix $P$ such that

$A=P^{-1}BP$

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

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

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

$=detB$

Thus, $det\left(A\right)=0\iff det\left(B\right)=0anddet\left(A\right)=1\iff det\left(B\right)=1$

Hence, option A.