課程名稱︰管理數學 課程性質︰必修 課程教師︰謝承熹 開課學院：管理學院 開課系所︰財金系 考試日期（年月日）︰97/11/10 考試時限（分鐘）：160分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Ⅰ.(49 points) Answer each of the following as true(T) or false(F). Justify your answer ,or you cannot get any point. Moreover, please give your answers in order. Thanks! 1.The set of all n×n and idempotent matrices is not a subspace of Mnn, where Mnn is the set of all n×n matrices under usual operations of matrix addition and scalar multiplication. 2.(Continue) The set of all n×n matrices with zero traceis not a subspace of Mnn. 3.Suppose that S={v1,...,vm} is a set of vector in R^n. If m>n,then S is linear dependent. 4.If A is an n×n and upper triangular matrix in which each aii≠0, then column of A is linear independent. 5.For scalars α and a nonsingular n×n matrix A, adj(αA)=(α^n-1)adjA. 6.For each n×n matrix A, it is impossible to find a solution for n×n matrix X in the matrix equation AX-XA=In. (Hint: Use the definition of trace.) 7.If A is a square matrix such that I-A is nonsingular, then A[(I-A)^(-1)]=[(I- A)^(-1)]A. (Hint: Consider A(I-A)=(I-A)A.) Ⅱ.(16 points) ╭ ╮ │1 0 0 1/3 1/3 1/3│ │0 1 0 1/3 1/3 1/3│ │0 0 1 1/3 1/3 1/3│ For the matrix A = │0 0 0 1/3 1/3 1/3│, Find A^300.(Hint: Use block │0 0 0 1/3 1/3 1/3│ │0 0 0 1/3 1/3 1/3│ ╰ ╯ multiplication and the definition of idempotent.) Ⅲ.(10 points) Construct a linear system of 3 equations in 4 unknowns that has x= ╭ ╮ ╭ ╮ ╭ ╮ │1│ │2│ │-3│ │0│+x2│1│+x4│ 0│ as its general solution. │1│ │0│ │ 2│ │0│ │0│ │ 1│ ╰ ╯ ╰ ╯ ╰ ╯ Ⅳ.(25 points) ╭ ╮ │1 X1 … X1 ^(n-2)│ │1 X2 … X2 ^(n-2)│ n-1 You are given det│: : … : │= Π(Xj-Xi). By applying induction │1 Xn-1… Xn-1^(n-2)│ j>i ╰ ╯ ╭ ╮ │1 X1…X1^(n-1)│ │1 X2…X2^(n-1)│ n method, please prove that det│: : … : │= Π(Xj-Xi). (Hint: Note that │1 Xn…Xn^(n-1)│ j>i ╰ ╯ n-2 Xn^(n-1)-Xi^(n-1)=(Xn-Xi)[Σ Xn^(n-2-k)．Xi^k] for i<n.) k=0 --

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