江西财经大学国际学院线性代数试题2009 Linear Algebra Test A

江西财经大学

2009－2010学年第一学期期末考试试卷

试卷代码：12063A 授课课时：48

课程名称：Linear Algebra 适用对象：2008级国际学院

1. Filling in t he Blanks (3’×6=18’)

(1) If ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=300210432A , then det(adj(A))= . (2) If ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=2100

110000010010A , then the inverse 1-A = (3)If ⎥⎥⎥⎥⎦

⎤⎢⎢⎢⎢⎣⎡=00000000b a a b b a a b A , then det(A)= (4) Let A be (4×4) matrix, and -1,2,4,6 are the eigenvalues of A . Then the eigenvalues of A -1 are .

(5) Let 10912,713αβ⎡⎤⎡⎤⎢⎥⎢⎥=-=-⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦

. Then the tripe products )(βαα⨯⋅= .

(6) If the rank of ⎥⎥⎥⎦

⎤

⎢⎢⎢⎣⎡-=11312211a A r(A)=2, then parameter a = . 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 be (3×3) inverse matrices, then ( ) is not always correct.

(A) T T T A B AB =)( (B) 111)(---=A B AB

(C) )()()(A adj B adj AB adj = (D) 222)(A B AB =

(2) Let A and B be (n ×n ) matrices , and 0,0≠=B AB , then . (A)0≠B (B) 0)(≠B adj (C) 0=T A (D) 222)(B A B A +=-

(3) If βα, are n dimension column vectors, and βα, are orthogonal, then ( ) is not always correct.

(A)0)(=⋅βα (B)0=βαT (C) βα, are linear independent. (D)0=αβ

(4) Let A be (m n ⨯) matrix ，and n m <, then the statement ( ) is always true.

(A) 0=AX has infinitely many solutions.

(B) b AX = has infinitely many solutions.

(C) 0=AX has no solutions.

(D)b AX = has no solutions.

(5) If n n ⨯ matrices A and B are similar, then the statement ( ) is always true.

(A) A,B have the same eigenvalues and eigenvectors.

(B) A,B only have the same eigenvectors.

(C) The rank of matrices A,B, such that r(A)=r(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 a basis of R 3. (D) 0321=A A A

3. (12’) If the system of linear equations is ⎪⎩⎪⎨⎧=+++=+-=++0)1(3112321

32321x a x x x ax x x x , then what value of a 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’) Let 33)(⨯=ij a A matrix such that det(A)=3, and let ij A denote the ij th cofactor of A. If

⎥⎥⎥⎦

⎤⎢⎢⎢⎣⎡=132333122232

112131

A A A A A A A A A

B , then calculate AB. 5. (10’) Suppose A is a (2×2) matrix such that 032=-+I A A , and ⎥⎦⎤

⎢⎣⎡=⎥⎦⎤⎢⎣⎡=31,12u where Au . Find u A and u A 32. 6. (15’) Find the all eigenvalues and all eigenvectors of matrix A, where ⎥⎥⎥⎦

⎤⎢⎢⎢⎣⎡----=020212022A .