国际学院2011年线代期末试卷

江西财经大学
11－12第一学期期末考试试卷
试卷代码：12063A 考试时间 110分钟 授课课时：48
课程名称：Linear Algebra （主干课程） 适用对象：2010级国际学院
1. Filling in t he Blanks (3’×6=18’)
(1) If ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=30
00320023404321A , then det (adj(A ))= . (2) If ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡-=212
313
1211
1143
21A , then =+++44434241A A A A . (3)If ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡--=1110161011A , and ⎥⎥⎥⎦
⎤⎢⎢⎢⎣⎡-=150401821B , then =T AB . (4) Let A be (4×4) matrix, and -1,2,3,6 are the eigenvalues of A . Then the eigenvalues of A -1 are .
(5) Let ⎥⎥⎥⎦
⎤⎢⎢⎢⎣⎡-=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-=222,104βα. Then the tripe products )(βαα⨯⋅= . (6) If ⎥⎥⎥⎦
⎤⎢⎢⎢⎣⎡--=11334221t A and B is a nonzero matrix, AB =0, then t = . 2. There are four choices in each question, but only one is correct. You should choose the correct one into the blank. (3’×6=18’)
(1) Let A and B are (3×3) invertible matrices, then ( ) is not always correct.
(A) T T T A B AB =)( (B) 111)(---=A B AB
(C) T T A A )()(11--= (D) 222)(A B AB =
(2) Let A and B be (n ×n ) matrices, then ( )
(A) AB =0⇒A =0 or B =0 (B) AB ≠0⇒A ≠0 and B ≠0
(C) AB =0⇒|A|=0 or |B|=0 (D) AB ≠0⇒|A|≠0 and |B|≠0
(3) Which of the following is incorrect? ( ).
(A) Let },,,{21p v v v be a set of vectors in m R , if m p >, then this set is linearly dependent.
(B) Let },,,{21p v v v be a set of vectors in m R , if m p ≤, then this set is linearly independent.
(C) Let A be (n n ⨯) matrix. The equation b Ax = has a unique solution for every (1⨯n ) column vector b if det(A )0≠.
(D) Let A be (n n ⨯) matrix. The equation b Ax = has a unique solution for every (1⨯n ) column vector b if A is nonsingular.
(4) Let A be (n n ⨯) matrix. Which of the following is not equivalent to the A has an inverse.
(A) A is nonsingular.
(B) A is row equivalent to I .
(C) the column vectors of A are linearly independent.
(D) 0=AX has infinitely many solutions.
(5) If n n ⨯ matrices A and B are similar, then the statement ( ) is always true.
(A) A and B have the same eigenvalues and eigenvectors.
(B) A and B only have the same eigenvectors.
(C) det( A )=det( B )
(D) The column vectors of A and B are all linear independent.
(6) If A is an (3×3) orthogonal matrix, ],,[321A A A A =, i A is the column vectors of A , then the statement ( ) is not always true.
(A) },,{321A A A is an orthogonal set.
(B) },,{321A A A is a linear independent set.
(C) },,{321A A A is an orthonormal basis of R 3.
(D) 1)det(=A
3. (12’) If the system of linear equations is ⎪⎩⎪⎨⎧-=++-=++-=++223321
321321kx x x x kx x k x x kx , then what value of k will make
the system has only solution, infinitely many solutions, no solutions, and when the system has infinitely many solutions, find its all solutions.
4. (12’) Suppose AXB =C , and ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-=021230201A ,⎥⎦⎤⎢⎣⎡=2513B , ⎥⎥⎥⎦
⎤⎢⎢⎢⎣⎡=312013C .Find matrix X. 5. (10’) Let W be a subspace, and let S be a spanning set for W . Find a basis for W , and calculate dim(W ) for set S.
⎪⎭
⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡--⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-=012,101,321,211S 6. (15’) Find all the eigenvalues and all eigenvectors of matrix A , where ⎥⎥⎥⎦
⎤⎢⎢⎢⎣⎡=122212221A , and find an orthogonal matrix Q such that AQ Q D T =, where D is diagonal.
7. (10’) There is a linear independent vectors set },,{321ααα, where
⎥⎥⎥⎦
⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡--=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=110,101,111321ααα. Please generate an orthonormal set },,{321u u u from vectors set },,{321ααα.
8. (5’) Suppose that },,{321v v v is a linear independent subset of m R . Show that the set },,{321211v v v v v v +++is also linearly independent.。