推 GI9:我不知道 我也沒有正確解答... 12/17 14:57
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※ 引述《GI9 ( )》之銘言：
: [a+b ab 0 0]
: [ 1 a+b ab 0]
: [ 0 1 a+b ab]
: [ 0 0 1 a+b]
: =M
: Show that if a and b are real numbers , which are not both zero ,
: then M is invertible
A4=|M|
A3=|a+b ab 0 |
| 1 a+b ab |
| 0 1 a+b|
A2=|a+b ab|
| 1 a+b|
A1=a+b A2=(a+b)^2-ab
An=(a+b)An-1-abAn-2
r^2-(a+b)r+ab=0
(r-b)(r-a)=0
if a=/=b
An=c1a^n+c2b^n
A1=a+b=c1a+c2b
A2=a^2+ab+b^2=c1a^2+c2b^2 ,a^2+ab+b^2=c1a^2+c2b^2
a^2+ab =c1a^2+c2ab ab+b^2=c1ab+c2b^2
-------------------------- ,------------------------
b^2=c2(b^2-ab) a^2=c1(a^2-ab)
c2=b/(b-a)
c1=a/(a-b)
1 1
An=-----a^n+1 + -----b^n+1
a-b b-a
If a=b
An=c1na^n+c2a^n
A1=2a = c1a +c2a
A2=3a^2=2c1a^2 +c2a^2
c1+c2=2
2c1+c2=3
c1=1 c2=1
An=(n+1)a^n
=>for a=/=b,
1 1
An=-----a^n+1 + -----b^n+1
a-b b-a
for a=b ,
An=(n+1)a^n
=>
a=b case ,ab=/=0 det不為零
a=/=b case,a^n+1-b^n+1=0 det為零
for n=4 a^5-b^5=0 a=be^i2nπ/5 n=0,1,2,3,4
a=b不合 其它的都是複數根了
不過感覺怪怪的XD,是這樣嘛?= =
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