Define a relation $\mathrm{R}$ over a class of $\mathrm{n} \times \mathrm{n}$ real matrices $\mathrm{A}$ and $\mathrm{B}$ as "ARB iff there exists a non-singular matrix $P$ such that $P A P^{-1}=B^{\prime \prime}$. Then which of the following is true?
Correct Option: , 3
$\mathrm{A}$ and $B$ are matrices of $n \times n$ order \& ARB iff there exists a non singular matrix $\mathrm{P}(\operatorname{det}(\mathrm{P}) \neq 0)$ such that $\mathrm{PAP}^{-1}=\mathrm{B}$
For reflexive $\mathrm{ARA} \Rightarrow \mathrm{PAP}^{-1}=\mathrm{A} \quad \ldots(1)$ must be true for $\mathrm{P}=\mathrm{I}$, Eq. (1) is true so ' $\mathrm{R}$ ' is reflexive For symmetric
$\mathrm{ARB} \Leftrightarrow \mathrm{PAP}^{-1}=\mathrm{B} \quad \ldots(1)$ is true
for $\mathrm{BRA}$ iff $\mathrm{PBP}^{-1}=\mathrm{A} \ldots(2)$ must be true $\because \mathrm{PAP}^{-1}=\mathrm{B}$
$\mathrm{P}^{-1} \mathrm{PAP}^{-1}=\mathrm{P}^{-1} \mathrm{~B}$
$\mathrm{IAP}^{-1} \mathrm{P}=\mathrm{P}^{-1} \mathrm{BP}$
$A=P^{-1} B P$
from (2) $\backslash$ (3) $\mathrm{PBP}^{-1}=\mathrm{P}^{-1} \mathrm{BP}$
can be true some $\mathrm{P}=\mathrm{P}^{-1} \Rightarrow \mathrm{P}^{2}=\mathrm{I}(\operatorname{det}(\mathrm{P}) \neq 0)$
So ' $R$ ' is symmetric For trnasitive
$\mathrm{ARB} \Leftrightarrow \mathrm{PAP}^{-1}=\mathrm{B} \ldots$ is true
$\mathrm{BRC} \Leftrightarrow \mathrm{PBP}^{-1}=\mathrm{C} \ldots$ is true
now $\operatorname{PPAP}^{-1} \mathrm{P}^{-1}=\mathrm{C}$
$\mathrm{P}^{2} \mathrm{~A}\left(\mathrm{P}^{2}\right)^{-1}=\mathrm{C} \Rightarrow \mathrm{ARC}$
So 'R' is transitive relation
$\Rightarrow$ Hence $\mathrm{R}$ is equivalence
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