let-T-be-a-n-n-matrix-with-integral-entries-and-Q-T-1-2-I-where-I-denote-the-n-n-identity-matrix-then-prove-that-matrix-Q-is-invertible- Tinku Tara June 10, 2024 Algebra 0 Comments FacebookTweetPin Question Number 208292 by universe last updated on 10/Jun/24 letTbean×nmatrixwithintegralentriesandQ=T+12IwhereIdenotethen×nidentitymatrixthenprovethatmatrixQisinvertible Answered by Berbere last updated on 10/Jun/24 LetT∈Mn(Z)⇒Q∈GLn(R)⇒Det(Q)≠0⇒T+12Iisinjective⇒Ker(T+I2)=0⇒∀x∈Rn−0Rn⇒T(x)+x2≠0Rn⇒−12∈C−{Spect(T)}soitsufficientToshowThat−12isnotarootofχTcaracteristicPolynomialofTsinceT∈Mn(Z)χT=Xn+∑n−1k=0akXk;∀k∈[0,n−1]ak∈ZSuposeThat1p;pprimeisrootofχ⇒1+∑n−1k=0pn−kak=0p∣1absurdp>2⇒1pcantbeerootofχ⇒−12∉spect(T)⇒T+12.InisinjecrivesinceT∈Mn(Z)⇒issurjectivebydim(ker)+dim(Im(T))=n⇒T+In2isbijective⇒QisinversiblinMn(Q)notMn(Z)QrationalnumberQissmallestFiledthatcontainZexempleQ=(321132);Q−=45(32−1−132)=(65−45−4565) Commented by Philton last updated on 13/Jun/24 −PhilipOnyeaka:Aperfectsolution,sir. Commented by Berbere last updated on 16/Jun/24 ThankYou Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Question-208277Next Next post: If-cos-cos-1-5-sin-sin-1-2-Find-cos- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.