作者ruby791104 (阿年：）)
看板Grad-ProbAsk
标题[理工] [线代]-四题行列式证明题
时间Thu Dec 10 01:29:33 2009
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
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