26. Suppose A is a 4x4 real matrix and y,y.y,y, are four lincarly independent vectors in R⁴. If Av₁=v₁+v₂, Av₂=v₂+v₃，Av₃=v₃+v₄, Av₄=v₁+v₄,then how many real eigenvalues does A have?
(A) 0
(B) 1
(C) 2
(D) 3
(E)4 

27. Suppose A is a 2x3 real matrix and B is a 3x2 real matrix., then which of the following matrix can be the product of BA?(A)(B)(C)(D)(E)

28. Suppose denote the standard vector norm in R2and R3. What is the smallest possible value for | As:ll if lI x=1? (A) (B) (C) 3 (D) 4(E)

29. Suppos denote the standard vector 110 nomm in R3. What is the smallest possible value for ||x||if Ax=b? (A) 1 (B) (C)(D) 2 (E)

30. Suppose A is a 3✖3 invertible matrix and I3 is the 3✖3 identity matrix. The followings are 5 statements about A that may or may not be true. (1) If p(x) is the characteristic polynomial of A, then the constant term of p(x) cannot be 0.(2) If a 3✖3 matrix B satisties AB+BA=0,then B is singular.(3) For any 3✖5 matrix C, rank(AC)= rank( C ).(4) If a 3✖3 matrix D is diagonalizable,then AD is also diagonalizable.(5)If E is a 3x3 matrix and AE is singular,then E is also singular.Which of the above statements about A is False? (A)(1) (B) (2) (C)(3) (D)(4) (E) (5)

31. If ||u||= 8 and ||v|| = 3, then what are the largest possible values of the inner product (u + v,v)? (A) 24 (B) 12 (C) 15(D)33(E) 45

32. Consider the system of linear equations:For what value of C will the system be singular? (A)-10 (B)-6 (C) -3 (D) 4 (E) 10

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)

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