Linear Algebra

Topic: Cramer’s Rule 333x − y = 4 −2x + y = −2(a)(b)−x + 2y = −7 x − 2y = −22 Use Cramer’s Rule to solve this system for z.3 Prove Cramer’s Rule.2x + y + z = 13x + z = 4x − y − z = 24 Here is an alternative proof of Cramer’s Rule that doesn’t overtly contain anygeometry. Write X i for the identity matrix with column i replaced by the vector ⃗xof unknowns x 1 , . . . , x n .(a) Observe that AX i = B i . (b) Take the determinant of both sides.5 Suppose that a linear system has as many equations as unknowns, that all ofits coefficients and constants are integers, and that its matrix of coefficients hasdeterminant 1. Prove that the entries in the solution are all integers. (Remark.This is often used to invent linear systems for exercises. If an instructor makesthe linear system with this property then the solution is not some disagreeablefraction.)6 Use Cramer’s Rule to give a formula for the solution of a two equations/twounknowns linear system.7 Can Cramer’s Rule tell the difference between a system with no solutions and onewith infinitely many?8 The first picture in this Topic (the one that doesn’t use determinants) shows aunique solution case. Produce a similar picture for the case of infinitely manysolutions, and the case of no solutions.