• # 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.

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