臺灣綜合大學系統107學年度學士班轉學生聯合招生考試
科目名稱:工程數學
類組代碼:D09


解答:(a)A=[010100001]⇒det(A−λI)=0⇒|−λ101−λ0001−λ|=−(λ−1)2(λ+1)=0⇒A的特徵值=1,−1(b)λ1=1⇒(A−λ1I)X=0⇒[−1101−10000][x1x2x3]=0⇒x1=x2⇒取v1=[1/√21/√20],v2=[001]λ2=−1⇒(A−λ2I)X=0⇒[110110002][x1x2x3]=0⇒{x1=−x2x3=0⇒取v3=[1/√2−1/√20]因此單範正交特徵向量(orthonormal eigen vector)為[1/√21/√20],[001],[1/√2−1/√20](c)A=[010100001]=[10−1101010][10001000−1][10−1101010]−1=P[10001000−1]P−1⇒sin(A)=P[sin(1)000sin(1)000−sin(1)]P−1⇒sin(A3)=P[sin3(1)000sin3(1)000−sin3(1)]P−1⇒sin(3A)=3sin(A)−4sin3(A)=P[3sin(1)−4sin3(1)0003sin(1)−4sin3(1)000−3sin(1)+4sin3(1)]P−1=[10−1101010][3sin(1)−4sin3(1)0003sin(1)−4sin3(1)000−3sin(1)+4sin3(1)][1/21/20001−1/21/20]=[3sin(1)−4sin3(1)03sin(1)−4sin3(1)3sin(1)−4sin3(1)0−3sin(1)+4sin3(1)03sin(1)−4sin3(1)0][1/21/20001−1/21/20]=[06sin(1)−8sin3(1)06sin(1)−8sin3(1)00003sin(1)−4sin3(1)]

解答:∬
沒有留言:
張貼留言