課程名稱︰線性代數 二
課程性質︰必修
課程教師︰黃漢水
開課學院：理
開課系所︰數學
考試日期（年月日）︰2009/4/22
考試時限（分鐘）： 10:20 ~ 13:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Let R be the field of all real numbers.
一 Let H = [ 1 1 -1 1 ] be in Mat4*4(R). Find det(H). (15%) [1]
[ 1 1 -1 -1 ]
[ 1 -1 -1 1 ]
[ 1 -1 1 -1 ]
二 Let A = [ a ][ 1 2 3 4 ] be in Mat4*4(R). Find the determinant det(A+I_4).
[ b ]
[ c ]
[ d ]
(20%)
三 Let A = [ 2 -2 2 ] be in Mat3*3(R). (25%)
[-2 5 -4 ]
[ 2 -4 5 ]
(1) Find the charateristic polynomial p(t) = (-1)^3 det(A-tI_3) of A.
(2) Find the minimal polynomial m(t) of A.
(3) FInd an invertible matrix K in Mat3*3(R) and a diagonal matrix
Λ in Mat3*3(R) such that AK = KΛ and (K^T)K = I_3.
(if such matrices exist)
(4) How many matrices B in Mat3*3(R) such that B^2 = A.
∞
四 Let u = {y_n} in R^∞ such that y_0 = 19, y_1 = -5
n=0
and for any n, 2y_n+2 = y_n+1 + y_n. (20%)
(1) Find y_n.
(2) Find the values y_10, y_13.
(3) Find the limit lim y_n. (if such limit exists)
n→∞
五 Let A = [ 0.2 0.2 0.2 ] in Mat3*3(R) and u = [ 20 ]. (20%)
[ 0.6 0.4 0.5 ] [ 20 ]
[ 0.2 0.4 0.3 ] [ 15 ]
(1) Find the all eigenvalues of A. [2]
(2) Find the lim A^n u. (if such limit exists)
n→∞
註解
[1] 所有「屬於」符號皆以英文單字 "in" 表示。
[2] "Find the all eigenvalues of A" 為題目原文，並非筆誤。
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※ 編輯: robertshih 來自: 140.112.91.80 (04/22 18:24)