推 ruby791104:iyenn大：謝謝你又幫我解決難題喔，下次也請多多關照！ 12/10 23:06
※ 引述《ruby791104 (阿年：）)》之銘言：
: 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.找出並證明三線共點的判別式(二維)。
: 以上，麻煩好心的大大們！（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
三線共點=>x,y有解
B=[b1 b2] r屬於Col(B)
let A=[b1 b2 r]
det(A)=0為判別式
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◆ From: 123.193.214.165
※ 編輯: iyenn 來自: 123.193.214.165 (12/10 02:50)