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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?
- a)R is symmetric, transitive but not reflexive.
- b)R is reflexive, symmetric but not transitive.
- c)R is an equivalence relation.
- d)R is reflexive, transitive but not symmetric.
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Define a relation R over a class of n × n real matrices A and B ...
A and B are matrices of n × n order & ARB iff there exists a non singular matrix P(det(P) ≠ 0) such that PAP-1 = B.
For reflexive:
ARA ⇒ PAP-1 = A ... (1) must be true for P = I, Eq.(1) is true, so 'R' is reflexive.
For symmetric:
ARB ⇔ PAP-1 = B ... (1) is true for BRA iff PBP-1 = A ...(2) must be true
PAP-1 = B
P-1PAP-1 = P-1B
IAP-1P = P-1BP
A = P-1BP ...(3)
From (2) & (3), PBP-1 = P-1BP can be true some P = P-1 ⇒ P2 = I (det(P) ≠ 0)
So, 'R' is symmetric.
For transitive:
ARB ⇔ PAP-1 = B... is true
BRC ⇔ PBP-1 = C... is true
Now, PPAP-1P-1 = C
P2A(P2)-1 = C ⇒ ARC
So, 'R' is a transitive relation.
⇒ Hence, R is equivalence.
For reflexive:
ARA ⇒ PAP-1 = A ... (1) must be true for P = I, Eq.(1) is true, so 'R' is reflexive.
For symmetric:
ARB ⇔ PAP-1 = B ... (1) is true for BRA iff PBP-1 = A ...(2) must be true
PAP-1 = BP-1PAP-1 = P-1B
IAP-1P = P-1BP
A = P-1BP ...(3)
From (2) & (3), PBP-1 = P-1BP can be true some P = P-1 ⇒ P2 = I (det(P) ≠ 0)
So, 'R' is symmetric.
For transitive:
ARB ⇔ PAP-1 = B... is true
BRC ⇔ PBP-1 = C... is true
Now, PPAP-1P-1 = C
P2A(P2)-1 = C ⇒ ARC
So, 'R' is a transitive relation.
⇒ Hence, R is equivalence.
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Community Answer
Define a relation R over a class of n × n real matrices A and B ...
Explanation:
Reflexivity:
- For a relation to be reflexive, every element in the set must be related to itself.
- In this case, for any matrix A, we can choose P as the identity matrix to satisfy the condition PAP^-1 = A.
- Therefore, every matrix A is related to itself, making the relation reflexive.
Symmetry:
- For a relation to be symmetric, if A is related to B, then B must be related to A.
- If A is related to B (A related to B means there exists a non-singular matrix P such that PAP^-1 = B), then we can choose P^-1 to satisfy the condition P^-1BP = A, which means B is related to A.
- Hence, the relation is symmetric.
Transitivity:
- For a relation to be transitive, if A is related to B and B is related to C, then A must be related to C.
- If A is related to B, there exists a non-singular matrix P1 such that P1AP1^-1 = B.
- If B is related to C, there exists a non-singular matrix P2 such that P2BP2^-1 = C.
- We can combine these two equations to get P2P1AP1^-1P2^-1 = C, which means A is related to C.
- Therefore, the relation is transitive.
Since the relation R satisfies the properties of reflexivity, symmetry, and transitivity, it is an equivalence relation. So, the correct answer is option 'C'.
Reflexivity:
- For a relation to be reflexive, every element in the set must be related to itself.
- In this case, for any matrix A, we can choose P as the identity matrix to satisfy the condition PAP^-1 = A.
- Therefore, every matrix A is related to itself, making the relation reflexive.
Symmetry:
- For a relation to be symmetric, if A is related to B, then B must be related to A.
- If A is related to B (A related to B means there exists a non-singular matrix P such that PAP^-1 = B), then we can choose P^-1 to satisfy the condition P^-1BP = A, which means B is related to A.
- Hence, the relation is symmetric.
Transitivity:
- For a relation to be transitive, if A is related to B and B is related to C, then A must be related to C.
- If A is related to B, there exists a non-singular matrix P1 such that P1AP1^-1 = B.
- If B is related to C, there exists a non-singular matrix P2 such that P2BP2^-1 = C.
- We can combine these two equations to get P2P1AP1^-1P2^-1 = C, which means A is related to C.
- Therefore, the relation is transitive.
Since the relation R satisfies the properties of reflexivity, symmetry, and transitivity, it is an equivalence relation. So, the correct answer is option 'C'.
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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?a)R is symmetric, transitive but not reflexive.b)R is reflexive, symmetric but not transitive.c)R is an equivalence relation.d)R is reflexive, transitive but not symmetric.Correct answer is option 'C'. Can you explain this answer?
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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?a)R is symmetric, transitive but not reflexive.b)R is reflexive, symmetric but not transitive.c)R is an equivalence relation.d)R is reflexive, transitive but not symmetric.Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 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?a)R is symmetric, transitive but not reflexive.b)R is reflexive, symmetric but not transitive.c)R is an equivalence relation.d)R is reflexive, transitive but not symmetric.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?a)R is symmetric, transitive but not reflexive.b)R is reflexive, symmetric but not transitive.c)R is an equivalence relation.d)R is reflexive, transitive but not symmetric.Correct answer is option 'C'. Can you explain this answer?.
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?a)R is symmetric, transitive but not reflexive.b)R is reflexive, symmetric but not transitive.c)R is an equivalence relation.d)R is reflexive, transitive but not symmetric.Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 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?a)R is symmetric, transitive but not reflexive.b)R is reflexive, symmetric but not transitive.c)R is an equivalence relation.d)R is reflexive, transitive but not symmetric.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?a)R is symmetric, transitive but not reflexive.b)R is reflexive, symmetric but not transitive.c)R is an equivalence relation.d)R is reflexive, transitive but not symmetric.Correct answer is option 'C'. Can you explain this answer?.
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