[2) Given the matrix A = 1 3 [a) Find the characteristic polynomial and check that the eigenvalues are A, = 1 y X, = 2. )which one of them has multiplicity? (b) Determine the corresponding diagonal matrix and the matrix change of basis. (c) Compute A. (It is enough to write the result as a product of three matrices. You do not have to perform additional computations)(1) Consider the following system of linear equations, + 2v -f 3 donde a, b E R son parametros. (a) Classify the system according to the values of the parameters a, b. [b) Solve the above system for the values of a =0 y b= 1.(1) Consider the following system of linear equations c+ by + z =1 y + 0z =b x+ (b - lly+ 2: =1 where a and bare parameters. [a) Determine, according to the values of a and b, whether the system is compatible, incompatible, determi nate or indeterminate. b) Solve the system for the values of a and b for which the system is compatible indeterminate. 2) Given the matrix [a) Compute the characteristic polynomial and the eigenvalues. b) For the value a =0, check that the matrix A is diagonalizable and find the matrix change of basis. (c) Study for which values of a 7 0, the matrix A is diagonalizable. (3) Given the vectors {(-1, 2, 1, -3), (3, -1, 1, -2), (2, 1, 2, -5), (2, -4, -2, 6]]in R'. (a) Compute the dimension of the linear subspace generated by them. b) Obtain the linear equations that determine the linear subspace generated by them. (4) Let A = {(x, y) ( R' : x 20, a + [w| $1). [a) Draw the set A. (b ) Draw the boundary, the interior and the closure of the set A. Is A closed, open, convex, bounded or compact ? (c) Consider the function f (x, y) = 1-2 If a # 1, If = = 1. Determine if the function / attains a maximum or a minimum in the set A. (5) Consider the function f : R' - R f(x, v) = if ( x, y) = (0,0), if (x, y) # (0, 0). (a) Study if the function f is continuous at the point (0, 0)). b) Determine whether the partial derivatives of f at the point (0, 0) ) exist. (c) Study if the function f is differentiable at the point (0, 0)). (6) Consider the function f(x, y) = +aly' + y' -3x -8y, (a) Compute the Taylor polynomial of order two of f around the point (0, 0). b) Determine if the function f is concave or convex in R" (7) Given the function f(x, y) = y' x - ay' - ax? (a) Show that (0, 0) is a critical point of / for any value of a and study its type of critical, according to the values of the parameter a. b) For a = 1 find all the critical points of f. c) For a = 1 classify the critical points of f obtained part b). (8) Consider the function /(x, y, 2) = x + y + z defined on the set A = ((x, y, =) ( R" : a' + 2y' + 4:' = 1}. (a) Find the Lagrange equations that determine the extreme points of f on A. b) Determine the points that satisfy the Lagrange equations and classify all the extreme points of f on A