JEE Main & Advanced AIEEE Paper (Held On 11 May 2011)

  • question_answer
    Consider the following relation R on the set of real square matrices of order 3. \[R=\{(A,B)|A={{P}^{-1}}BP\]for some invertible matrix P}. Statement -1 : R is equivalence relation. Statement - 2 : For any two invertible \[3\times 3\] matrices M and N,\[{{(MN)}^{-1}}={{N}^{-1}}{{M}^{-1}}.\]     AIEEE  Solved  Paper (Held On 11 May  2011)

    A)  Statement-1 is true, statement-2 is a correct explanation for statement-1.

    B)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

    C)  Statement-1 is true, Statement-2 is false.

    D)  Statement-1 is false, Statement-2 is true.

    Correct Answer: B

    Solution :

                    for reflexive \[(A,A)\in R\]\[\Rightarrow \]\[A={{P}^{-1}}AP\]which ....... for P = I \[\therefore \]reflexive for symmetry As \[(A,B)\in R\] for matrix P \[A={{P}^{-1}}BP\] \[\Rightarrow \]\[PA{{P}^{-1}}=B\] \[\Rightarrow \]\[B=PA{{P}^{-1}}\] \[\Rightarrow \]\[B=({{P}^{-1}})A({{P}^{-1}})\] \[\therefore \]\[(B,A)\in R\]for matrix \[{{P}^{-1}}\] \[\therefore \]R is symmetric for transitivity \[A={{P}^{-1}}BP\] and    \[B={{P}^{-1}}CP\] \[\Rightarrow \]\[A={{P}^{-1}}({{P}^{-1}}CP)P\] \[\Rightarrow \]\[A={{({{P}^{-1}})}^{2}}C{{P}^{2}}\] \[\Rightarrow \]\[A={{({{P}^{2}})}^{-1}}C({{P}^{2}})\] \[\therefore \] \[(A,C)\in R\]for matrix \[{{P}^{2}}\] \[\therefore \]R is transitive so R is equivalence

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