33. Given the row-reduced echelon form of a matrix A. What is the closest vector to b = [0,3,0, 1]^T in the row space of A?
(A) [1,1,2,0]^T
(B) [0,-2,-1,1]^T
(C) [-1, 2,0, -1]^T
(D) [-1,2,1,0]^T
(E) [1,1,3,1]^T

34. Let ， which of the following statement is incorrect? (A) The rank of A is 3. (B) The inverse matrix is (C) If we define the characteristic polynomial of A is f(t)=det(A-tIn), then f(1)=-t3+3t2+3t-1. (D) One of the eigenvalues is -1 and the corresponding eigenvector is (-2, 1, 2)T. (E) The matrix cannot be diagonalized over the real field.

35. An n x n skew-symmetric matrix A is defined as At= -A. Please find the dimension of this matrix. (A)n (B) n2 (C) n(n + 1)/2 (D)n(n-1)/2 (E) n +1 

36. A vector space is spanned by {1,cos(t),sin(t)} for -π≤t≤π . If a vector: v=a.1+b.sin(t)+cㆍcos(t) is the closest vector in this vector space to a continuous function: f(t)=t for -π ≤t≤π, what is this closest v? The inner product of two continuous functions: fand g within [a, b] is defined by You may need the following integral:. (A)v=1+2sin(t)-2cos(t) (B) v=1-2cos(t) (C) v= 2sin(r) (D) v=1+2sin(t) (E) v=-2cos(t)

37. A quadratic equation is described as: x2 +8xy + 7y2 = 225. Which of the following statement is incorrect? (A) The curve is centered at the origin (B) This quadratic curve is an ellipse (C) One of the principal axes is (D) The other principal axis is (E) The shortest distance from this quadratic curve to the origin is 5

38. Which of the following statements is true? (A) Let A be an arbitrary matrix of size m X n. If m ＞ n, then rank( A ) ＜ rank(At). (B) Let A be an arbitrary matrix of size m ✖n. If m ＞ n,then rank( A ) ＞ rank(At). (C) If W1, W2 are two subspaces of V, then the union W1U W2 is also a subspace of V. (D) Let W be a subspace of a vector space V. Then, dim(W) ≤ dim(V). (E) Let A be a square matrix, k be a constant. Then, det(kA) = kdet( A ).

39. Consider the matrix . How many different mattices are there such that B2 = A? (A) 0 (B) 1 (C) 2 (D) 4 (E) 16

40. Consider the Fibonacci series a0 = 1, a1 = 2,and an = an-1 + an-2 for any n≥ 2. Therefore, we have {an} = {1,2,3,5,8, ...}. The general expression for an as a function of n can be found via the following method； first, definc , and an define the state vector . Thus, we have the initial condition ，and  the relation Vn+1 = Avn. The following general expression can be derived: an = are the eigenvalues of A. How many ofthe following statements are true? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

151. The following information is given about a group of high-risk borrowers.i) Of all these borrowers, 30% defaulted on at least one student loan.ii) Of the borrowers who defaulted on at least one car loan, 40% defaulted on at least one student loan.iii) Of the borrowers who did not default on any student loans, 28% defaulted on at least one car loan.A statistician randomly selects a borrower from this group and observes that the selected borrower defaulted on at least one student loan.Calculate the probability that the selected borrower defaulted on at least one car loan.(A) 0.33(B) 0.40(C) 0.44(D) 0.65(E) 0.72

152. An insurance company issues policies covering damage to automobiles. The amount of damage is modeled by a uniform distribution on [0, b]. The policy payout is subject to a deductible of b/10. A policyholder experiences automobile damage. Calculate the ratio of the standard deviation of the policy payout to the standard deviation of the amount of the damage. (A) 0.8100 (B) 0.9000 (C) 0.9477 (D) 0.9487 (E) 0.9735

153. A policyholder purchases automobile insurance for two years. Define the following events:  F = the policyholder has exactly one accident in year one.  G = the policyholder has one or more accidents in year two. Define the following events: i) The policyholder has exactly one accident in year one and has more than one accident in year two. ii) The policyholder has at least two accidents during the two-year period. iii) The policyholder has exactly one accident in year one and has at least one accident in year two. iv) The policyholder has exactly one accident in year one and has a total of two or more accidents in the two-year period. v) The policyholder has exactly one accident in year one and has more accidents in year two than in year one. Determine the number of events from the above list of five that are the same as F ∩ G. (A) None (B) Exactly one (C) Exactly two (D) Exactly three (E) All

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