Advanced Calculus fi..

34 Advanced Calculus, Fifth Edition2. Evaluate the determinant as in Gaussian elimination:3. Solve by Gauss-Jordan elimination:4. Find the inverse matrix:5. In the following systems the elimination process shows that the determinant of coefficientsis 0. Carry out the process and find all solutions:a) 2x-y+z=3,~+2y-z=1,5~-5y+4~=86. Consider two equations in three unknowns:a) Show that if Gaussian elimination can be carried out to solve for x and y, thenby writing z = t the solutions become parametric equations for a line in space(Section 1.3).b) Assume that the two equations represent two planes in space. Interpret geometricallythe case in which the equations have no solution and the case in which eliminationleads to a second equation 0 = 0.7. Consider four equations in three unknowns:Assume that the pair of equations for i = 1,2 and the pair for i = 3,4 each representsa line in space as in Problem 6(a). Thus the solutions of the four equations represent thepoints common to two lines in space. Discuss the geometrical alternatives that can occurand relate them to the set of solutions of the given equations.8. Let A be an m x n matrix. Let C = (cij) be the m x m matrix which differs from the identityonly in that chh = A # 0. Let B be the m x m matrix which differs from the identityonly in the kth column, in which bkk = 1.a) Show that CA is obtained from A by multiplying the hth row by A.b) Show that BA is obtained from A by adding bjk times the kth row to the jth row forall j except k.C) Show that B and C are nonsingular.