課程名稱︰線性代數二
課程性質︰數學系必修
課程教師︰黃漢水
開課學院：理學院
開課系所︰數學系
考試日期︰2009年04月22日
考試時限：110分鐘，10:20-13:10
是否需發放獎勵金：是
試題 :
Let R be the field of all real numbers.
一 Let H = ┌ 1 1 -1 1 ┐ ∈ Mat4*4(R). Find det(H). (15%)
｜ 1 1 -1 -1 ｜
｜ 1 -1 -1 1 ｜
└ 1 -1 1 -1 ┘
二 Let A = ┌ a ┐[ 1 2 3 4 ] ∈ Mat4*4(R). Find the determinant det(A+I_4).
｜ b ｜
｜ c ｜
└ d ┘
(20%)
三 Let A = ┌ 2 -2 2 ┐ ∈ 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 ∈ 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 ┐ ∈ 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. [1]
(2) Find the lim A^n u. (if such limit exists)
n→∞
註解
[1] "Find the all eigenvalues of A" 為題目原文，並非筆誤。
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