1條pure determinant

case 1: abc=0 (let a=0)
LH=det( -bc bc+b^2 bc+c^2// 0 0 c^2 //0 b^2 0)
=(bc)^3=RH
case 2: abc does not equal 0
Multipliy a,b,c to the 1st, 2nd, 3rd rows respectively, then
LH=1/(abc)*
丨-abc abc+ab^2 abc+ac^2丨
丨bca+ba^2 -cba cba+bc^2丨
丨cab+ca^2 abc+cb^2 -abc 丨
the 1st, 2nd, 3rd column of the above det have common factor a, b, c respectively, then
LH=
丨-bc ac+ab ab+ac丨
丨bc+ab -ca ab+bc丨
丨bc+ca ac+bc -ab |
R1+R2+R3, then
LH=
| ab+bc+ca ab+bc+ca ab+bc+ca |
| bc+ab -ca ab+bc |
| bc+ca bc+ca -ab |
then
LH=(ab+bc+ca)*
| 1 1 1 |
| bc+ab -ca ab+bc |
| bc+ca bc+ca -ab |
then
LH=(ab+bc+ca)*
| 1 0 0 |
| bc+ab -ab-bc-ca 0 |
| bc+ca 0 -ab-bc-ca |
then LH=(ab+bc+ca)^3


圖片參考：http://imgcld.yimg.com/8/n/AD04686329/o/701006060116813873419860.jpg


2010-06-07 03:09:43 補充：
To:Nelson
抱歉啦!

就當是參考吧:
http://img683.imageshack.us/img683/4390/20279512.png