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© University of London 2016 UL16/0434 Page 1 of 6 D1~~MT1173_ZA_2016_d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON MT1173 ZBBSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas in Economics and Social Sciences AlgebraTuesday, 10 May 2016 : 10:00 to 13:00 Candidates should answer all FIVEquestions. All questions carry equal marks. Calculators may not be used for this paper. PLEASE TURN OVER
1. (a) Show that the line1given byxyz=-101+t-111, tR,is parallel to the plane with Cartesian equation 2x-y+3z=0.Find the equation of the line through the point (-1,0,1) which isperpendicular to the plane. Then find the point where this perpendicular lineintersects the plane. Hence, or otherwise, find the distance of the line1to theplane.(b) LetAbe anm×nmatrix andBann×nmatrix. Simplify, as much aspossible, the expression(ATA)-1AT(B-1AT)TBT(ATA)-1ATassuming that any matrix inverse in the expression is defined.Define what is meant by therankof anm×nmatrixA.IfAis anm×nmatrix and (ATA)-1exists, prove that rank(A)=n.(c) A geometric progressionyt,t0, has fourth termy3=1 and sixth termy5=1/4. Show that there are two possible values of the common ratioxandfind the corresponding values of the first terma. Then find the correspondingvalue of the sum to infinity of each progression.PLEASE TURN OVERUL16/0434Page 2 of 6
2. (a) Using row operations to put the augmented matrix into reduced row echelon

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