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主題課程_行列式和線性方程式:行列式#107854 

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1.

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.



2.

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



3.

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.



4.

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.



5.

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) =det det .

(D) det = det where A and B are row equivalent matrices.

(E)



6.

Let A, B ∈ , where F = C or R. Which of the following statements are true?

(A) tr(AB)= tr(BA)

(B)tr(AB)=tr tr

(C) = tr

(D)

(E)tr(A±B)=tr ±tr



7.

Let A, B ∈ , where F = C or R. Which of the following statements are true?

(A) det (AB) = det(BA)

(B) det(AB) = det det

(C) det(B-1AB)= det

(D) det = (det )'

(E)det(A±B)=det +det



【非選題】
8.
If there are two nxn matrices A, B and one scalar r,

【題組】
(A) det
(A) = det
(B) implies A = B.

(B) det(rA+B) = rdet
(A)+det
(B).

(C) det(AB)T =det
(A)det
(B).

(D) det
(A) = det
(B) where A and B are row equivalent matrices.

(E) det(AB)-) -det(A')det(B' l).


【非選題】
9.【已刪除】【題組】

【非選題】
10.【已刪除】【題組】

【非選題】
11.【已刪除】【題組】

【非選題】
12.

 Let 627e07e563535.jpgCompute the following determinants: 


【題組】

(a) det627e08161890c.jpg.



【非選題】
13.【題組】

(b) det627e089e15a8e.jpg.



【非選題】
14.【題組】

(c) det627e08b797e85.jpg.



【非選題】
15.【題組】

(d) 627e08d1e9430.jpg



【非選題】
16.

Let A, B, C, and D be nxn matrices with A invertible. Prove that det 627e0928e2bbc.jpg (det A) det(627e093e3eafc.jpg) .



【非選題】
17.

Given a n X n tridiagonal matrix A as below:
627e0958e9fd7.jpg
Please find the determinant of A when n = 2020.



【非選題】
18.
Prove or disprove the following statements:

【題組】(a) There exist nxn real matrices A and B such that AB-BA=I, where I is then nxn identity matrix. 


【非選題】
19.【題組】(b) If A and Bare nxn matrices and det(A-B)=0,then det
(A)=det
(B).


【非選題】
20.【題組】

(c) If A is a nilpotent matrix (namely, A is an nxn matrix such that 627e09a399f1e.jpg = O for some positive integer k), then det(I + A)=1 , where I is the nxn identity matrix.



【非選題】
21.

Find the values of x such that the given matrix is not invertible. 627e09ceba1b2.jpg



【非選題】
22.

Compute the determinant of 627e09e67272e.jpg



【非選題】
23.

Let B be the n x n matrix, compute det
(B).627e0a3c3c708.jpg



【非選題】
24.

Given two matrices M and N, please calculate
627e0a547ec0c.jpg


【題組】(a)  det(M)


【非選題】
25.【題組】(b)det(MN)

【非選題】
26.【題組】(c)det(adj(M), where adj(M) satisfies M -adj(M) = det (M) .I

【非選題】
27.
4.Let A and B be 3 x3 matrices with det
(A) = 5, det
(B) = 10, and det(A+B) = 60. Decide the following values.

【題組】(a) det(A+A).


【非選題】
28.【題組】(b)  det(A2B+AB2).

【非選題】
29.【題組】

If 627e12b1c94c6.jpg find det(K). 



【非選題】
30.

Let 627e133a9e221.jpg and a,b ∈ 627e1355ef97d.jpg. Find det
(A). 



【非選題】
31.

Solve for t if det 627e13750b135.jpg=0



【非選題】
32.

Let 627e1394d75d3.jpg and 627e13a398daf.jpg be the nxn identity matrix. Find det(A +627e13b65379a.jpg.). 



【非選題】
33. Please find the determinant of A. 
627e13da44b57.jpg


【非選題】
34.

Prove that det(M)= 627e13f6218dd.jpg for the following matrix: 

627e140f3e172.jpg



【非選題】
35.

Suppose that Px(xt) is a polynomial of order & with leading coefficients, ax, k = 0,ㆍㆍ.,n -1.
That is, 627e1498c248c.jpg+...+ a1x+ a0, k = 0,.n -1. Then
627e14374f7ea.jpg=_______



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主題課程_行列式和線性方程式:行列式#107854-阿摩線上測驗

主題課程_行列式和線性方程式:行列式#107854