A. MillerM340 Exam 2April 7, 1993There is a total of 100 points. Put your answers on separatesheets of paper. Keep this exam, hand in the answers and all worksheets.1. (8 points) Find all eigenvalues ofA =3 −4 00 3 20 0 0?2. (8 points) One eigenvalue ofA =2 −2 01 −3 01 −1 4is 4. Find an eigenvector of A (other than the zero vector) associated withthe eigenvalue 4.3. (8 points) Note that A = P DP−1where:A ="1 2−1 4#P ="2 11 1#D ="2 00 3#P−1="1 −1−1 2#What is A100?4. (8 points) Give an example of a matrix which is not similar to a diagonalmatrix and prove that it is not.15. (12 points) Find the unique solution to the system of differential equa-tions:dx1dt= 3x1− 4x2dx2dt= 2x1− 3x2x1(0) = 1, x2(0) = 2Note:"3 −42 −3#="2 11 1#"1 00 −1#"1 −1−1 2#6. (10 points) For A an m × n matrix the null space of A, N (A), is definedbyN (A) = {X ∈ Rn×1: AX = Zm×1}.Prove that N (A) is a subspace of the vector space, Rn×1, of all n × 1 columnmatrices.7. (10 points) For each of the following sets of vectors prove they are linearlyindependent or prove they are linearly dependent.(a) In the vectorspace of polynomials, the set of three vectorsx2+ x, x2− x, x3.(b) In the vectorspace R3×1the set of three vectors1−10,101,0−1−128. (12 points) For v1, v2, . . . , vnvectors in a vector space V define(a) span{v1, v2, . . . , vn},(b) v1, v2, . . . , vnare linearly independent,(c) v1, v2, . . . , vnare linearly dependent, and(d) v1, v2, . . . , vnare a basis for V .9. (9 points) Prove that if v1, v2, . . . , vnare linearly dependent, then forsome k with 1 ≤ k ≤ n we have vk∈span{v1, v2, . . . vk−1, vk+1, . . . , vn}. Givea complete proof without referring to any other Theorem that may have beenproven in class.10. (15 points) Suppose we classify the women in a country according as towhether they live in an urban (U) or rural (R) area. Suppose each woman hasjust one daughter, who in turn has just one dauughter, and so on. Supposefurther that(a) for urban women, 10% of the daughters settle in rural areas and 90%settle in an urban area, and(b) for rural women, 50% of the daughters settle in rural areas and 50%settle in an urban area.1. Give the transition matrix for this Markov chain.2. Let p be the probability that the greatgranddaughter of an urbanwoman is a rural women. (woman, daughter, granddaughter, great-granddaughter) What entry of what matrix is p? What is p?3. After many generations have passed what percentage of women areurban and what percentage are