12. Let A be a 2 x 2 matix defined s.Which of the tolowing statements are true? (A) The characteristic function is (B) The eigenvalues are (C) 'The eigenvectors are (1,i) and (1, -i). (D) The eigenvectors are (-1,i) and (-1, -i). (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.

12. Let A be a 2 x 2 matix defined s.Which of the tolowing statements are true?
(A) The characteristic function is
(B) The eigenvalues are
(C) 'The eigenvectors are (1,i) and (1, -i).
(D) The eigenvectors are (-1,i) and (-1, -i).
(E) None of the above.

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