26. What is the determinant of the following matrix?


(A) el - kf

(B) 0

(C) bcgiy - bdghy

(D) adfyh + bcxdk – cazej

(E) bcy +gi-bd - aghy

27. Let L(x) = Ax represent a linear transformation from R² to R², where:A =What kind of geometry operation is L?(A) L is the linear operator that rotates each x in R² by 45° in the clockwise direction(B) L is the linear operator that reflects each vector x in R² about the x1 axis and then rotates it 90° in the counterclockwise direction(C) L doubles the length of x and then rotates it 30° in the counterclockwise direction(D) L reflects each vector x about the line x2 = x1 and then projects it onto the x1-axis(E) L rotates x 270° in the counterclockwise direction

28. Consider the system of linear equations:x + y + az = bx - 2y + z = 1x - 8y - 11z = -7For what values of a and b will the system have infinitely many solutions?(A) a = 7, b = 1(B) a = 11, b = 7(C) a = 11, b = 5(D) a = 5, b = 7(E) a = 7, b = 5

29. Let {v₁, v₂, v₃} = {[1, 1, 0]ᵀ, [0, 1, 1]ᵀ, [1, 0, 1]ᵀ} and {u₁, u₂, u₃} = {[1, 2, 4]ᵀ, [1, 3, 5]ᵀ, [1, 1, 1]ᵀ} be two ordered bases for the vector space R³. If x = v₁ - 2v₂ + 3v₃, what is the coordinates of x with respect to the basis {u₁, u₂, u₃}?(A) [7, -6, 3]ᵀ(B) [1, -2, 3]ᵀ(C) [4, -1, 1]ᵀ(D) [-2, 0, 3]ᵀ(E) [2, -3, 0]ᵀ

30. For the matrixwhat is the corresponding eigenvalue with the multiplicity 2?(A) 1(B) 2(C) 3(D) 4(E) 5

31. Consider {0} and {1} as the basis for a vector space in C², then the natural basis for the tensor product space C² ⊗ C² has the ordered bases {00}, {10}, {01}, {11}. The tensor product of two vectors is defined from their decomposition on the bases.What is the corresponding matrix formed bywheredenotes the transpose.(A)(B)(C)(D)(E)

32. Let R⁴ˣ⁴ denote the collection of all 4 × 4 real matrices. Define the lineartransformation T: R⁴ˣ⁴ → R⁴ˣ⁴ by T( A ) = Aᵀ + 2A for any 4 × 4 matrix A.What is the rank of T?(A) 4(B) 8(C) 10(D) 12(E) 16

33. Let R⁶ denote the 6-dimensional real vector space and ||•|| denote the standard vector norm in R⁶. How many 6 × 6 real matrices U satisfy || Ux || = || x || for all x ∈ R⁶ and |trace(U)| = 6?(A) 1(B) 2(C) 3(D) 6(E) Infinitely many

34. Let Rθ denote the rotation matrix through an angle 0＜θ＜counterclockwise about the origin in R². Let e₁ = [1,0], e₂ = [0,1]ᵀ be the standard basis of R². Suppose A is a 2x2 matrix satisfying Ae₁ = Rθe₁ and RθAe₂ = e₂. The followings are 5 statements about A that may or may not be true.(1) A²e₁ = R₂θe₁(2) det( A ) = 1(3) trace( A ) = 2cos θ(4) A is a positive definite matrix(5) ARθ = RθAWhich of the following combination of statements about A are all true?(A) (1)(2)(B) (1)(3)(C) (2)(3)(D) (3)(4)(E) (4)(5)

35. Given a matrix Find the eigenva lue of A².(A) 1 and 2(B) -1 and -2(C) 2 and 4(D) 1 and 8(E) 1 and 16

36. What are the dimensions of the row space, column space, and null space for the matrix?(A) 3, 5, 5(B) 3, 5, 2(C) 3, 3, 2(D) 2, 3, 3(E) 2, 2, 3

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