作者hiakira (哈罗小明)
看板NTU-Exam
标题[试题] 97上 谢承熹 管理数学 期中考
时间Sat Nov 22 17:39:09 2008
课程名称︰管理数学
课程性质︰必修
课程教师︰谢承熹
开课学院:管理学院
开课系所︰财金系
考试日期(年月日)︰97/11/10
考试时限(分钟):160分钟
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
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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