無年度 - 主題課程_行列式和線性方程式：行列式#107854

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 its
diagonal 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.

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

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.

Denote det A as the determinant of the matrix A, and denote as the inverse of the
matrix A. Let A, B, and P be square matrices. Which of the following statements
is/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.

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

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

(a) det.

(b) det.

(c) det.

(d)

Let A, B, C, and D be nxn matrices with A invertible. Prove that det (det A) det() .

Given a n X n tridiagonal matrix A as below:

Please find the determinant of A when n = 2020.

(c) If A is a nilpotent matrix (namely, A is an nxn matrix such that = O for some positive integer k), then det(I + A)=1 , where I is the nxn identity matrix.

Find the values of x such that the given matrix is not invertible. 

Compute the determinant of 

Let B be the n x n matrix, compute det(B).

If find det(K). 

Let and a,b ∈ . Find det(A). 

Solve for t if det =0

Let and be the nxn identity matrix. Find det(A +.). 

Prove that det(M)= for the following matrix: 

Suppose that Px(xt) is a polynomial of order & with leading coefficients, ax, k = 0,ㆍㆍ.,n -1.
That is, +...+ a₁x+ a₀, k = 0,.n -1. Then
=_______