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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