Menu Close

Q1-Let-M-2-be-the-set-of-square-matrices-of-order-2-over-the-real-number-system-and-R-A-B-M-2-M-2-A-P-T-BP-for-some-non-singular-P-M-Then-R-is-A-symmetric-B-transit




Question Number 119797 by Ar Brandon last updated on 27/Oct/20
Q1  Let M_2  be the set of square matrices of order 2 over  the real number system and       R={(A,B)∈M_2 ×M_2 ∣A=P^( T) BP  for some                 non-singular P ∈M}  Then R is  (A) symmetric  (B) transitive  (C) reflexive on M_2   (D) not an equivalence relation on M_2     Q2  For any integer n, let I_n  be the interval (n, n+1).  Define         R={(x, y)∈R∣both x, y ∈ I_n  for some n∈Z}  Then R is  (A) reflexive on R  (B) symmetric  (C) transitive  (D) an equivalence relation
$$\mathrm{Q1} \\ $$$$\mathrm{Let}\:{M}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{square}\:\mathrm{matrices}\:\mathrm{of}\:\mathrm{order}\:\mathrm{2}\:\mathrm{over} \\ $$$$\mathrm{the}\:\mathrm{real}\:\mathrm{number}\:\mathrm{system}\:\mathrm{and} \\ $$$$\:\:\:\:\:\mathcal{R}=\left\{\left({A},{B}\right)\in{M}_{\mathrm{2}} ×{M}_{\mathrm{2}} \mid{A}={P}^{\:\mathrm{T}} {BP}\:\:\mathrm{for}\:\mathrm{some}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{non}-\mathrm{singular}\:{P}\:\in{M}\right\} \\ $$$$\mathrm{Then}\:\mathcal{R}\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{symmetric} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{transitive} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{reflexive}\:\mathrm{on}\:{M}_{\mathrm{2}} \\ $$$$\left(\mathrm{D}\right)\:\mathrm{not}\:\mathrm{an}\:\mathrm{equivalence}\:\mathrm{relation}\:\mathrm{on}\:{M}_{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Q2} \\ $$$$\mathrm{For}\:\mathrm{any}\:\mathrm{integer}\:{n},\:\mathrm{let}\:{I}_{{n}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{interval}\:\left({n},\:{n}+\mathrm{1}\right). \\ $$$$\mathrm{Define} \\ $$$$\:\:\:\:\:\:\:\mathcal{R}=\left\{\left(\mathrm{x},\:\mathrm{y}\right)\in\mathbb{R}\mid\mathrm{both}\:\mathrm{x},\:\mathrm{y}\:\in\:{I}_{{n}} \:\mathrm{for}\:\mathrm{some}\:{n}\in\mathbb{Z}\right\} \\ $$$$\mathrm{Then}\:\mathcal{R}\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{reflexive}\:\mathrm{on}\:\mathbb{R} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{symmetric} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{transitive} \\ $$$$\left(\mathrm{D}\right)\:\mathrm{an}\:\mathrm{equivalence}\:\mathrm{relation} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *