Find the determinant for the following matrix
(A)-5
(B) 5
(C)1
(D)-1
(E) 0 

複選題Which of the following statements about the determinant is true?(A) For any square matrix A, det = -det A.(B) The determinant of the n x n identity matrix is .(C) 'The deterrninant of an upper triangular mnatrix is always equal to the product of itsdiagonal entries.(D) Performing a row addition operation on a square matrix does not change its determinant.(E) Performing a scaling operation on a square matrix does not change its determinant.

複選題Given which of the following statements is/are true? (A) det =det =0. (B) det = -40. (C) det = det. (D) det = 20. (E) None of the above are true.

複選題Denote det A as the determinant of the matrix A, and denote as the inverse of thematrix A. Let A, B, and P be square matrices. Which of the following statementsis/are true?(A) It is always true that det AB = det BA.(B) If the columns of A are linearly dependent, then det A = 0.(C)It is always true that det (A + B) = det A + det B.(D)If A is invertible, then det (E) Suppose that Pis invertible. Then det = det A.

複選題If there are two nxn matrices A, B and one scalar r,(A) det = det implies A = B.(B) det(rA+B) = rdet+det.(C) =detdet.(D) det = det where A and B are row equivalent matrices.(E)

複選題Let A, B ∈ , where F = C or R. Which of the following statements are true?(A) tr(AB)= tr(BA)(B)tr(AB)=trtr(C) = tr(D) (E)tr(A±B)=tr±tr

複選題Let A, B ∈ , where F = C or R. Which of the following statements are true?(A) det (AB) = det(BA)(B) det(AB) = detdet(C) det(B-1AB)= det(D) det = (det )'(E)det(A±B)=det+det

Let A be an m x n matrix whose null space has dimension k. Which conclusion is correct? (A) The dimension of Null(AT) is k. (B) The dimension of row space of A is m-k. (C) The dimension of column space of A is m-k. (D) The dimension of row space of A'is n-k. (E) The dimension of column space of A is n-k.

Let A =xyT, where t and y are two nonzero vectors of Rn, n ＞ 1. Which of the following statements is/are true? (A) rank( A ) = 1 and the range space of A is Spanf{y}. (B) rullity( A ) = 2 and the null space of A is Spanfe,{x，y}. (C) Trace( A ) = 1 and det( A ) =0 (D) A is always diagonalizable. (E) None of the above.

.(5%) Determine the rank of the matrix in the following.  (A)0 (B)1 (C)2 (D)3 (E) 4

11. (10%) Let Q be an n✖ n matrix. Then which of the following set is not a subspace? (A) Col O (B) Null Q (C) rank Q (D) Row Q (E) None of the above

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