[理工] [線代]交大100

[death888 (ZZZZ)]

9.Suppose A:3x3 has three distinct eigenvalues 0,1,3 with corresponding

eigenvectors u,v,w. 何者敘述正確

A.The rank of A must be 2
B.The matrix A^2-I is invertible
C.v and w must span the column space of A
D.The least squares error of Ax=2v+3w+u must be ∥u∥^2


10.Suppose A:3x3 = xy^T is rank 1 matrix.何者敘述正確

A.The eigenvector matrix S must be invertible
B.A must have one non-zero eigenvalue
C.When A is facotrized by SVD into UΣV^T,the only non-zero singular value
  is ∥x∥∥y∥
D.When A is factorized by SVD into UΣV^T,U must include x/∥x∥ as
  one of its column vectors


請問這兩題要怎麼選 腦袋轉不過來

覺得觀念考好多orz

順便再問幾個選項

1. A,B,A+B,都是n*n可逆矩陣, then A^-1 +B^-1 也可逆

我是選FALSE 總覺得舉得出反例 不過想不到
         ┌     ┐
2.If     │a b c│
      det│d e f│=7
         │1 3 5│
         └     ┘
第三列改1 0 1的時候 det為9,   then 第三列改1 6 9的時候=5
(排版麻煩 我用文字敘述)

true or false?

我對題目給的兩種展開 然後把三種coefficient用變數A B C代換

但這樣也只有兩條方程式 三個未知數解不出來 有別的辦法嗎?
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◆ From: 111.250.214.190
※ 編輯: death888        來自: 111.250.214.190      (02/19 00:35)
推 annheilong:det那題我會 X=(bf-ce) Y=(af-cd) Z=(ae-bd)
→ annheilong:展開det得 X-3Y+5Z=7 X+Z=9 => X=9-Z, Y=1/3(4Z+2)
→ annheilong:代入1 6 9那條 X-6Y+9Z= 9-Z-2(4Z+2)+9Z = 5
推 juan19283746:線代好難ˊ ˋ
推 daniel770624:同一樓做法