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.

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(A^T).
(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.

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