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CAS EC 505 Mathematics for EconomicsExtra Midterm 12/16/18IWrite your answers in the blue booklets providedIPlease hand in the question sheet at the end of the examIScienti°c calculators are allowedIGood luck!1. Consider the following matrix:A=26642142114874393775:(a) Give the de°nition of the rank of a matrix and °ndrank(A)(b) How many solutions does the systemAx=bhave?(c) Find the solution(s) to the systemAx=bifbT= (5;8;5;8):2. Letx= (2;2;1)andy= (1;0; y3):(a) Findy3such that the angle°betweenxandysatis°escos°=13:(b) Find a point-normal expression for a plane that passes through the origin andcontains the vectorsxandy:(c) Does the pointp= (°1;1;1)belong to this plane?3. De°ne the matrixBand the vectorbas:B=°2113±andb=°12±:(a) Use the de°nition of linearity to verify whether the functionf(x) =Bx+bislinear or not.(b) Give the de°nition of a concave function. Isf(x) =Bx+bconcave?(c) Provide the de°nition of an inverse of a matrix and °nd²BTB³°1:1