課程名稱︰線性代數一
課程性質︰數學系必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期︰2007年01月10日
考試時限：110分鐘
是否需發放獎勵金：是
試題 :
Write your answer on the answer sheet. You should include in your answer every
piece of computations and every piece of reasonings so that corresponding
partial credit could be gained.
(1) Calculate the determinant of the matrix (30 points).
┌ 1 2 1 -1 2 -2 ┐
｜ 1 4 2 -1 0 0 ｜
A= ｜ 1 2 2 0 4 0 ｜,
｜ 2 4 2 -3 4 -2 ｜
｜ -1 -2 -1 1 0 0 ｜
└ 0 2 1 0 -2 0 ┘
(2) Let
A=┌ -8 5 ┐
└ -30 17 ┘,
(a) Determine the characteristic polynomial of A (10 points).
(b) Find the eigen-values of A. And for each eigen-value of A, find an
eigen-vector of A corresponding to this eigen-value (20 points).
(c) Find a matrix C such that A=C‧D‧C^(-1), where D is a diagonal matrix
(10 points).
(d) Show that if
A^n = ┌ a_n b_n ┐
└ c_n d_n ┘,
then lim -a_n = lim b_n = lim -c_n = lim d_n = ∞ (10 points).
n→∞ n→∞ n→∞ n→∞
(3) Find the area of the parallegram in R^3 determined by the vectors (1,1,0)
and (2,0,7) (10 points).
(4) Find a 2*2 matrix A such that 10 is the only eigen-value of it but the
corresponding eigen-space is only one-dimensional (10 points).
(試題結束)
P.S. 符號a_n表示a右下標n
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 203.73.252.117