複選題

9. Let A be an n x n matrix whose (i，j) component is aij. The trace of A is defined as trA = Let B and C be n x n matrices. Which of the following statements are true?
(A) tr(AB)= tr(BA).
(B) tr(ABC)=tr(CBA).
(C) tr(A^TB)=tr(AB^T).
(D) tr(A+B)=(trB)(trA).
(E) None of the above. 

1. Which of the following statements are true? (A) f is differentiable at zo implies that f is continuous at zo. (B) f(x) = x4 + 3 has an inflection point. (C)=1. (D) =0. (E) None of the above.

2. Let f(r) = In(1 + z). Which of the following statements are true? (A) The Taylor series about o = O js x-x2 + x3-x4+ .... for |x |＜1. (B) The Taylor series about t = O is x -x2/2 + x3/3-x4/4+..., for |x |＜1. (C) f'(x) is increasing in z for r ＞ -1. (D) f"(x) is increasing in a for x > -1. (E) None of the above.

3. Consider f(x) =1/ (1 +x). Which of the following statements are true? (A) The Taylor series about x = O is 1 +x +x2 +x3 +..., for  |x | ＜1. (B) The 'Taylor series aboutx = O is 1 -x+x2-x3+⋯...for |x | ＜1. (C) (D) (E) None of the above. 

複選題4. Which of the following statements are true? (A)(B)(C)(D)(E)

5. column vector. Which of the following statements are true? (A) The gradient of f with respect to a is (2x1, 2x2, 2x3)T. (B) The Hessian of f with respect to ar is (2,2,2)T. (C) The minimum value of f subject to 2x1 + 2x2 - 2x3 = 7 is 5. (D) The maximum value of f subject to 2x1 + 2x2 - x3 = 7 is 10. (E) None of the above.

6. Let f(x) = (sina)/x. Which of the following statements are true? (A) (B)(C)(D) maxxf(x)=1. (E) None of the above.

7.Let Which of the following statements are true? (A)(B) (C)(D) (E) None of the above.

複選題8. Let A be an m x n matrix. Which of the following statements are true? (A) nullity( A ) + rank( A )=n. (B) rank( A )=rank(AT). (C) If m = 7 and n = 5, then rank( A ) is at most 5. (D) Suppose that m = n. Then, "A is singular" "A has rank n". (E) None of the above.

複選題10. Let A be an n x n matrix whose (i, j) component is aij. Let f be a real-valued function defined on A. Let  ∇Af( A ) be the gradient of f( A ) with respect to A； ∇Af( A )is delined as an n x n matrix whose (i, j) entry is . Let B and C be n x n matrices. Which of the following statements are true? (A) If f( A ) = tr(AB), then ∇Af( A )= B. (B) If f( A ) = tr(AB), then ∇Af( A ) = BT. (C) If f( A ) = tr(AATC), then ∇Af( A ) = CA + CTAT. (D) If f( A ) == tr(AATC), then ∇Af(  A  ) = CA + CAT. (E) None of the above.

11. Which of the following statements are true? (A) is singular if and only if Ax = b has infinitely many solutions for every n x 1 vector b. (B) Every matrix transformnation is a linear transformation. (C) If A is a nonsingular upper triangular matrix, then the adjoint matrix is lower triangular. (D) The dimension of the zero space {0} is 0. (E) None of the above.

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