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Q2 15 Points Let A € Mnxn(R). Define trace(A) = {1=1 diji (1. e. the sum of the diagonal entries) and tr : Mnxn (R) +R, A H tQ2.4 3 Points Show that for any A € Mnxn (R) there is B e ker(tr) and a E R such that A = B+a. E for E E Mnxn (R) with (E)i,j

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Aus- trace (A)= a1 + 912e tain let A = [auj] E Mnxn(IR) Bi lai for til ai - Tela is o take Teace (A) Now a = big vid ity consI ari n Trace (A) n Trace (A)- Trace (A) = 0 6 B € Kerfile) Hence A = Bta. I where a= Trace (A) CS Scanned with CamScanner

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