EC 505 A Midterm1 Fall 2018.pdf - CAS EC 505 A1 Mathematics for Economics Midterm 1 I Write your answers in the blue booklets provided I Please hand in

EC 505 A Midterm1 Fall 2018.pdf - CAS EC 505 A1 Mathematics...

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CAS EC 505 A1 Mathematics for Economics Midterm 1 10/3/18 I Write your answers in the blue booklets provided I Please hand in the question sheet at the end of the exam I Scienti°c calculators are allowed I Good luck! 1.(a) Find the rank of the following matrix:A=241°21211°1117°5°135(b) Give the de°nition of an inverse of a matrix. DoesAhave an inverse? Why/whynot?(c) Leta; b; c2R:Solve the systemAx=dwheredT= (a; b; c):What must be truefora; b; cin order for there to be a solution?2.(a) Provide a de°nition of linear independence.(b) For what value(s) oftare the vectorsx; y;andzbelow linearly dependent?x=@1811A; y=@t°211A; z=@411A(c) For what values oftiscos°= 0;where°is the angle betweenxandy?3. LetBbe the matrix:B=24bb2°211°1bb°135whereb2R:(a) For what values ofbdoes the systemAx=chave a unique solution for allc?(b) State Cramer±s rule and use it to solve the system whenb= 1andcT= (4;2;4).(c) DoesBhave an inverse whenb=°1? 1