JEE Main 2021 · 18 March · evening · Single correct · +4 / −1
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 reflexive, transitive but not symmetric
- (b) R is symmetric, transitive but not reflexive.
- (c) R is reflexive, symmetric but not transitive
- (d) R is an equivalence relation
Answer and solution
Answer (d) R is an equivalence relation
Solution
Maths sir · 4 steps
For each property, just find the right P: the identity, then the inverse, then the product.
- Reflexive. Take P = I, which is non-singular. Then IAI⁻¹ = A, so ARA holds for every matrix A.
- Symmetric. Suppose PAP⁻¹ = B. Multiply on the left by P⁻¹ and on the right by P to get A = P⁻¹BP. With Q = P⁻¹, which is also non-singular, this reads A = QBQ⁻¹, so BRA.
- Transitive. Suppose B = PAP⁻¹ and C = QBQ⁻¹. Substitute the first into the second. The product QP is non-singular because |QP| = |Q| |P| ≠ 0, so ARC.
C = Q(PAP⁻¹)Q⁻¹ = (QP) A (QP)⁻¹
- So. R is reflexive, symmetric and transitive, so it is an equivalence relation, option (d).
The idea to keep. To test a relation defined by "there exists a matrix P", build the P each property needs. The identity, the inverse and the product of invertible matrices are all invertible.
Answer from the NTA final answer key. Solution by PrepWiser.