西南大学课程考核
《 线性代数 》课程试题 【A 】卷
3. A =⎪⎪⎪
⎭
⎫ ⎝⎛300020001, then A -1 is ( )
A. ⎪⎪⎪
⎪⎪⎪⎪
⎭
⎫ ⎝
⎛10002100031 B. ⎪⎪⎪⎪⎪
⎪⎭
⎫ ⎝
⎛310
00210001 C. ⎪⎪⎪⎪
⎪⎭
⎫ ⎝⎛21000100031 D.
⎪⎪⎪
⎪⎪⎪⎪
⎭
⎫ ⎝⎛10003100021
4. A , B , C are square matrices and AB =AC , we can conclude ( ):
A. A is a zero matrix
B. A is a zero matrix if B ≠C
C. B =C if A is not zero matrix
D. B =C if det A ≠0
5. Rewrite ⎪
⎪⎭⎫ ⎝⎛⎪⎪⎭⎫
⎝⎛-=2121211422
),(),(x x x x x x f as a standard quadratic form x T A x , A = ( ) A. ⎪
⎪⎭⎫ ⎝⎛1-332
B. ⎪⎪⎭⎫ ⎝⎛1-422
C. ⎪⎪⎭⎫
⎝⎛1-212 D. ⎪⎪⎭
⎫
⎝⎛1001
III. Solve the following problem (15points)
Given a homogenous linear system: ⎪⎩⎪
⎨⎧-=+++=+++=+++2
321
321321)1()1(0
)1(λλλλλx x x x x x x x x
Find the value of λ when the system has:
a) Exactly one solution; b) No solution;
c) Infinitely many solutions.
In case c), describe all possible solutions as a linear combination of some vectors. Answer:
西南大学课程考核(试题【A】卷) ——————————————
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《 线性代数 》课程试题 【A 】卷
IV . Calculate the determinant of matrix A (9points)
⎪⎪⎪⎪
⎪⎭
⎫
⎝
⎛--+---+---=11
1
1111111
1
1
1111x x x x A Answer:
西南大学课程考核(试题【A】卷)
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Matrix A and B satisfy AB=A+2B,
⎪
⎪
⎪
⎭
⎫
⎝
⎛
=
3
2
1
-
1
1
3
2
4
A, Calculate matrix B. Answer:
《线性代数》课程试题【A】卷
VI. (12points)
A linear transformation in 2 is defined as the composition of the following steps:
(1)Rotate θ counterclockwise (逆时针旋转θ), where θ=tan-1(0.75); (0.75的反正切)
(2)Scale by the factor 2 (放大到原来的2倍);
(3)Reflection through x1-axis (绕x1轴翻转).
Determine: the standard matrix of this linear transformation.
Is this linear transformation a one to one linear transformation? Explain the reason. Answer:
西南大学课程考核(试题【A】卷)
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线——————————————VII. System analysis (12points)
The Internet service supply is a duopoly (两家垄断) in a city. Investigation shows that in customer who currently contracted with company A, 10% will contract with company B in the next year. On the other hand, in the customer who currently contracted with company B, 15% will move to company A in the next year.
a)Establish the model which describes the relationship between the market share of current
year and the market share of the next year.
b)Estimate the vector which describes the long-term market share (长期以后的市场份额) of
these two companies. Explain why it is a stable status and how to converge to this status.
(解释为何此份额是稳定的，以及是怎样收敛到此数值的).
Answer:
《线性代数》课程试题【A】卷
VIII. Prove the following statement (7points)
A is a 3×3 matrix and A T=-A.
Prove: A is not invertible.
Answer:。