私立中原大學八十九學年度碩士班招生考試命題紙


所組別：資訊工程學系 科目：計算機數學 考試時間：4月29日 第2節

1. [10%] Find the number of integer solutions to the equation , where .

2. [15%] Solve the following system of recurrence relations
.

3. [10%] Consider nine nonnegative real numbers with sum 90, show that there must be four of the numbers having sum at least 40.

4. [15%] Let be the floor function of .
(a)[5%] Find all such that.
(b)[10%] Find a formula forwhere n is a positive integer.

5. 　 [25%] Let A = be the given matrix.
  1. [5%] Find its determinant.
  2. [10%] Find the null space of A.
  3. [10%] Find the kernel of A.

6. [25%] Given the matrix B = .
  1. [10%] Find all eigenvalues of B.
  2. [10%] Find the corresponding eigenvectors.
  3. [5%] Perform the similarity transformation for B.

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