课程名称︰数量方法入门 课程性质︰必修 开学前先修课 课程教师︰张胜凯 开课学院：社科学院 开课系所︰经研所 考试日期（年月日）︰99.09.03 考试时限（分钟）：110 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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