課程名稱︰數量方法入門
課程性質︰必修 開學前先修課
課程教師︰張勝凱
開課學院：社科學院
開課系所︰經研所
考試日期（年月日）︰99.09.03
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Problem 1. (15 points (5,5,5))
Write the following quadratic form in matrix notation with a symmetric
matrix:(a)x^2+y^2-xy (b)4x^2+5y^2+z^2+2xy+2yz (c)-x^2+2y^2+xy
Verify whether these forms are positive(or negative) denite,
semidenite, or indenite.
Problem 2. (15 points (4,3,3,5))
Let
[1 a 0]
A=[4 5 3]
[1 0 2]
(a) For which a is det(A) = 0? Show that the columns of A are
linearly dependent in that case.
(b) If we interchange the second and third row, show that det(A) = 0
changes sign.
(c) If we subtract 4 times the rst row from the second row, show
that det(A) = 0does not change.
(d) Obtain the inverse of matrix A for a = 1
Problem 3. (20 points (5,5,5,5))
Let i be a 4X1 vector of ones, that is, i=(1, 1, 1, 1)'
(a) Find the matrix M where M=I4-(1/4)ii' and I4 is 4X4 idebtity matrix.
(b) Show that M is symmtrix and idempotent.
(c) Let x=(x1,x2,x3,x4)'.Find Mx,
(d)Find Mi.
Problem 4. (30 points (15,15))
(a) Solve the following two dierence-equation system:
x(t+1)=-x(t)-2y(t)+24
y(t+1)=-2x(t)+2y(t)+9
with initial conditions x(0)=10 and y(0)=9
(b) Solve the following two dierential-equation system:
x'(t)=x(t)+12y(t)-60
y'(t)=-x(t)-6y(t)+36
with initial conditions x(0)=13 and y(0)=4
Problem 5. (20 points (5,5,5,5))
For matrix A
[1 3 0 -2 0]
A=[2 6 1 -1 0]
[1 2 1 1 1]
(a) Find the basis for row space of A, Row(A).
(b) Find the basis for column space of A, Col(A).
(c) Find the basis for Null space of A, Null(A).
(d) Find the rank of A.
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※ 編輯: sheng1300905 來自: 61.62.92.132 (09/04 14:40)