江西财经大学线性代数试题061110018369

2004-2005学年第一学期期末考试试卷

课程代码： 12063C 课时： 48 课程名称：Linear Algebra 线性代数(英) 适用对象：2003级国际学院

1. Filling in the blanks (3’×6=18’)

(1) Let A=[α,γ2,γ3,γ4]andB=[β,γ2,γ3,γ4] be (4×4) matrices, and det(A)=4, det(B)=1, then det(A+B)= . ⎡0⎢1⎢

(2) Let A=⎢0

⎢⎣0

100⎤000⎥⎥

010⎥, then the eigenvalues of A are . ⎥

001⎦

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

⎧⎧⎧⎪0⎫⎪⎪2⎫⎪⎪1−k⎫⎪

α1=⎨4⎬,α2=⎨3−k⎬,α3=⎨2⎬. Then the number k is .

⎪⎪⎪⎩2−k⎪⎭⎩1⎪⎭⎩3⎪⎭

(4) Let A and B be (n×n) matrices, and A2=A, B2=B, A+B=I, then AB+BA= .

(5) Let A and B be (3×3) matrices, and AB=2A+B, where

⎡202⎤

⎥-1B=⎢040⎥,then (A-I)= . ⎢

⎢⎦⎣202⎥

⎧⎧⎧⎪⎪0⎫⎪⎪2⎫⎪⎪1⎫

α=⎨−1⎬,β=⎨0⎬,γ=⎨2⎬. Then the scalar triple product

(6) Let

⎪⎪⎪⎭⎩1⎪⎭⎩1⎪⎭⎩0⎪

α⋅(β×γ)= .

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

(1) If A and B are symmetric (n×n) matrices, then AB is also symmetric.

( )

(2) A consistent (3×2) linear system of equations can never have a unique solution. ( ) (3) If u·v =0, then either u =0 or v =0 . ( ) (4) If x is an eigenvector for A, where A is nonsingular, then x is also an eigenvector for A-1. ( ) (5) If A, B, and C are (n×n) matrices such that AB=AC and det(A)≠0, then B=C. ( ) (6) If A is an (n×n) matrix such that det(A)=1, then Adj[Adj(A)]=A. ( ) 3. (15’)

111⎤⎡1+x

11⎥⎢11−x

11+y1⎥. Calculate the determinant of the matrix ⎢1

⎢111−y⎥⎣1⎦

4. (15’)

⎧x1+x2+kx3=4⎪2

xkxxk−++=23, determine Consider the system of equations ⎨1⎪⎩x1−x2+2x3=−4

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conditions on k 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’)

Suppose that {v1,v2,v3} is a linearly independent subset of Rm. Show that the set 6. (15’)

{v1,v1+v2,v1+v2+v3}is also linearly independent.

⎡101⎤

−1−1A=⎢020⎥ABA=BA+3I. Find B. and Let ⎢⎥⎣101⎦

7. (15’)

⎡2−1⎤100A=, calculate A. Let ⎢⎣−12⎥⎦

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