※ 引述《ruby791104 (阿年：）)》之铭言： : 1.Let A be an n ×n matrix and α a scalar. : Show that det(αA) = α^n det(A) : 2.Let A be a nonsingular matrix. : Show that det(A^-1) = 1/det(A) : 3.Consider the 3 ×3 Vandermonde matrix : ┌ ┐ : │ 2│ : │1 x x │ : │ 1 1│ : │ │ : │ 2│ : V = │1 x x │ : │ 2 2│ : │ │ : │ 2│ : │1 x x │ : │ 3 3│ : └ ┘ : (a)Show that det(V) = (x2 - x1)(x3 - x1)(x3 - x2). : [Hint:Make use of row operation Ⅲ.] : (b)What conditions must the scalars x1, x2, x3 : satisfy in order for V to be nonsingular? : 4.找出并证明三线共点的判别式(二维)。 : 以上，麻烦好心的大大们！（鞠躬 半夜被吵醒...冏很大ORZ 1. Ann=[a11 a12 ... a1n] [a21 a22 ... a2n] [... ... ... ...] [an1 an2 ... ann] kAnn=[ka11 ka12 ... ka1n] [ka21 ka22 ... ka2n] [... ... ... ... ] [kan1 kan2 ... kann] det(kAnn)=|ka11 ka12 ... ka1n| |ka21 ka22 ... ka2n| |... ... ... ... | |kan1 kan2 ... kann| =k|a11 ka12 ... ka1n| |a21 ka22 ... ka2n| |... ... ... ...| |an1 kan2 ... kann| =k^2|a11 a12 ... ka1n| |a21 a22 ... ka2n| |... ... ... ... | |an1 an2 ... kann| =...=k^ndet(A) 2.det(AA^-1)=det(A)det(A^-1)=det(I)=1 det(A^-1)=1/det(A) 3.V=[1 a a^2] [1 b b^2] [1 c c^2] det(V)=|1 a a^2|=|1 a a^2 | |1 b b^2| |0 b-a b^2-a^2 | |1 c c^2| |0 c-a c^2-a^2 | = | b-a b^2-a^2 | | c-a c^2-a^2 | =(b-a)(c+a)(c-a)-(c-a)(b-a)(b+a) =(c-a)(b-a)(c+a-b-a) =(c-a)(b-a)(c-b) a=/=c=/=b s.t det(A)=/=0 imply A is nonsingular 4. 令 a11x+a12y=a13 a21x+a22y=a23 a31x+a32y=a33 =>[a11 a12] [a13] [a21 a22][x]=[a23] [a31 a32][y] [a33] Bv=r 三线共点=>x,y有解 B=[b1 b2] r属於Col(B) let A=[b1 b2 r] det(A)=0为判别式 -- 为者常成.行者常至 --

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