江西财经大学线性代数试题061110238955

2005-2006学年第一学期期末考试试卷

课程代码： 12063A 课时： 48 课程名称：Linear Algebra 线性代数(英) 适用对象：2004级国际学院

1. Filling in the blanks (3’×6=18’) (1) Let

A=[2α,3γ]andB=[β,2γ] be (2×2) matrices, and det(A)=3,

det(B)=-2, then det(A+B)= .

⎡201⎤⎢320⎥−1A=(I+A)= . (2) Let ⎥, then ⎢⎢⎦⎣002⎥

⎡1⎤

⎢1⎥1α=⎢⎥

(3) Let 2⎢1⎥is a 4 dimension vector, A=

⎢⎥⎣1⎦

I−2ααT

and

B=I+2ααT, then the matrix AB= .

(4) Let A be a (3×3) matrix, and 1,2,3 are the eigenvalues of A. Then the eigenvalues of I+A* are .

(5) Let S={α1,α2,α3} be a linearly dependent set of vectors, where

⎡1⎤⎡1⎤⎡1⎤

⎥,α=⎢2⎥,α=⎢3⎥α1=⎢1⎢⎥2⎢⎥3⎢⎥. Then the scalar k is . ⎢⎢⎢⎦⎣k⎥⎦⎣3⎥⎦⎣1⎥

⎡1⎤⎡2⎤

⎢0⎥,β=⎢1⎥α=⎢⎥⎢⎥. Then the cross product α×β= . (6) Let

⎢⎢⎣3⎥⎦⎣1⎥⎦

1

2. Determining the following statement whether it is true(T) or false(F) (2’×6=12’)

(1) If A is an (m×n) matrix such that AX=0 for every X in Rn,then A is the (m×n) zero matrix. ( )

(2) If A and B are nonsingular (n×n) matrices then A－B is also nonsingular. ( )

(3) If U×V =0, then either U =0 or V =0 . ( )

(4) If A is nonsingular with A-1=AT, then det(A)=1 ( ) (5) If S is (n×n) and nonsingular, then A and S−1AShave the same eigenvalues. ( )

(6) If A is an (n×n) matrix such that det(A)=1,then Adj[Adj(A)]=A.

( )

3. (15’) Evaluate the determinant of the matrix

bbb⎤⎡a+b

⎢b⎥a+bbb⎢⎥⎢bba+bb⎥. ⎢⎥

bba+b⎦⎣b

4. (15’)

⎧2x1+λx2−x3=1

⎪

λx1−x2+x3=2, determine

Consider the system of equations ⎨

⎪4x+5x−5x=−1

23⎩1

2

conditions on λ that are necessary and sufficient for the system to be has only solution, infinite solutions, and no solution, and express the solutions by vectors. 5. (10’)

Let U and V be nonzero vectors such that U=V. show that U-V and U+V are orthogonal. 6. (15’)

⎡102⎤

⎥030A=⎢⎢⎥ and ABA−1=BA−1+3I. Find B. Let

⎢⎣201⎥⎦

7. (15’)

⎡1⎤⎡232⎤⎥⎢a42⎥1ξ=⎢A=⎢⎥ is an eigenvector of ⎢⎥, Let

⎢⎢⎣−1⎥⎦⎣1b1⎥⎦

(Ⅰ) Find the numbers a and b.

(Ⅱ) Find the eigenvalues of the matrix A.

3