Define a relation R over a class of n×n real matrices A and B as “ARB iff there exists a non-singular matrix P such that PAP−1=B ”.Then which of the following is true?

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Define a relation R over a class of $n \times n$ real matrices A and B as “ARB iff there exists a non-singular matrix P such that ${PAP}^{- 1} = B$ ”.Then which of the following is true?

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R is symmetric, transitive but not reflexive,

R is an equivalence relation

R is reflexive, symmetric but not transitive

R is reflexive, transitive but not symmetric

answer is C.

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Detailed Solution

${IAI}^{- 1} = A ∴ R is reflexive Let {IAI}^{- 1} = B \Rightarrow A = I^{- 1} BI \Rightarrow A = {IBI}^{- 1} ∴ R is symmetric Let {IAI}^{- 1} = B \Rightarrow A = B {IBI}^{- 1} = C \Rightarrow B = C ∴ A = C \Rightarrow {IAI}^{- 1} = C ∴ R is Transitive$