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(D) Let the relation be defined as $R = \{(A, B) \mid B = P^{-1} A P 	ext{ for some non-singular matrix } P\}$.For reflexivity: Since $A = I^{-1} A I

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an equivalence relation

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(D) Let the relation be defined as $R = \{(A, B) \mid B = P^{-1} A P 	ext{ for some non-singular matrix } P\}$.For reflexivity: Since $A = I^{-1} A I$ where $I$ is the identity matrix, $(A, A) \in R$. Thus, $R$ is reflexive.For symmetry: Let $(A, B) \in R$. Then $B = P^{-1} A P$. Multiplying by $P$ on the left and $P^{-1}$ on the right, we get $P B P^{-1} = A$. Let $Q = P^{-1}$. Then $A = Q^{-1} B Q$. Thus, $(B, A) \in R$. So, $R$ is symmetric.For transitivity: Let $(A, B) \in R$ and $(B, C) \in R$. Then $B = P^{-1} A P$ and $C = Q^{-1} B Q$ for some non-singular matrices $P$ and $Q$. Substituting $B$, we get $C = Q^{-1} (P^{-1} A P) Q = (P Q)^{-1} A (P Q)$. Since $PQ$ is non-singular, $(A, C) \in R$. Thus, $R$ is transitive.Since the relation is reflexive, symmetric, and transitive, it is an equivalence relation.

In the set of all $3 \times 3$ real matrices, a relation is defined as follows: $A$ matrix $A$ is related to a matrix $B$ if and only if there exists a non-singular $3 \times 3$ matrix $P$ such that $B = P^{-1} A P$. This relation is

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