作者pattrick (派特瑞克)
看板NTU-Exam
标题[试题] 96下 黄维信 线性代数 期末考
时间Sat Jun 21 12:24:48 2008
课程名称︰线性代数
课程性质︰选修
课程教师︰黄维信
开课学院:工学院
开课系所︰工程科学与海洋工程学系
考试日期(年月日)︰2008/6/20
考试时限(分钟):120
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
1.suppose the 4 by 4 matrix M has four equal rows all containing a,b,c,d.
we know that det(M)=0. the problem is to find det(I+M) by any method:
(10%)
| 1+a b c d |
| a 1+b c d |
det(I+M)=| |
| a b 1+c d |
| a b c 1+d |
2.True of false , with reason if true and counterexaple if false:(15%)
(a) if A and B are identical except bi1=2ail, then detB=2detA.
(b) the determinant is the product of the pivots.
(c) if A is invertible and B is singular, then A+B is invertible.
(d) if A is invertible and B is singular, then AB is singular.
(e) the determinant of AB-BA is zero.
3.True of false, with reason if true and counterexaple it false:(15%)
(a) for every matrix A, there is a solution to du/dt=Au starting from u(0)
=(1,...,1).
(b)every invertible matrix can be diagonalized.
(c)every diagonalizable matrix can be invertible.
(d)exchanging the rows of 2by 2 matrix reverses the signs of its eigenvalues
(e)if eigenvectors x and y corrspond to distinct eigenvalues,
H
then x y=0.
4.solve the second-order equation(20%)
2
d u ┌ ┐ ┌ ┐ , ┌ ┐
-----=│-5 -1│u u(0)=│1│ and u (0)=│0│.
2 │-1 -5│ │0│ │0│
d t └ ┘ └ ┘ └ ┘
5.show that A and B are similar by finding M so that B=M^(-1)AM:(15%)
┌1 0┐ ┌0 1┐
A=│ │ and B=│ │
└1 0┘ └0 1┘
6.show the condition that ax^2+2bxy+cy^2 is positive define. Decide whether F
=x^2*y^2-2x-2y has a minimum.(15%)
┌0 1 0┐
7.find the SVD and the pseudoinverse of └1 0 0┘ (20%)
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