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.

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.

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