課程名稱︰線性代數
課程性質︰選修
課程教師︰黃維信
開課學院：工學院
開課系所︰工程科學與海洋工程學系
考試日期（年月日）︰2008/6/20
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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