SOLUTION: <pre> 1 1 1 Show that x y z = (y - x)(z-x)(z-y) X2 y2 z2</pre>

Algebra ->  Matrices-and-determiminant -> SOLUTION: <pre> 1 1 1 Show that x y z = (y - x)(z-x)(z-y) X2 y2 z2</pre>       Log On


   

Question 977171: 
	        1	1       1
Show that       x	y       z      = (y - x)(z-x)(z-y)
	        X2     y2      z2
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
We are to show that:

abs%28matrix%283%2C3%2C1%2C1%2C1%2Cx%2Cy%2Cz%2Cx%5E2%2Cy%5E2%2Cz%5E2%29%29%22%22=%22%22%28y+-+x%29%28z-x%29%28z-y%29

Expand the determinant by the top row:

1%2Aabs%28matrix%282%2C2%2Cy%2Cz%2Cy%5E2%2Cz%5E2%29%29%22%22-%22%221%2Aabs%28matrix%282%2C2%2Cx%2Cz%2Cx%5E2%2Cz%5E2%29%29%22%22%2B%22%221%2Aabs%28matrix%282%2C2%2Cx%2Cy%2Cx%5E2%2Cy%5E2%29%29%22%22=%22%22

1%2A%28yz%5E2-y%5E2z%29-1%2A%28xz%5E2-x%5E2z%29%2B1%2A%28xy%5E2-x%5E2y%29

yz%5E2-y%5E2z-xz%5E2%2Bx%5E2z%2Bxy%5E2-x%5E2y

Let's multiply out the right side:

%28y-x%29%28z-x%29%28z-y%29%22%22=%22%22%28y-x%29%28z%5E2-yz-xz%2Bxy%29%22%22=%22%22
yz%5E2-y%5E2z-xyz%2Bxy%5E2+-+xz%5E2%2Bxyz%2Bx%5E2z-x%5E2y%22%22=%22%22%22%22=%22%22yz%5E2-y%5E2z%2Bxy%5E2+-+xz%5E2%2Bx%5E2z-x%5E2y

They are the same except for the order of the terms.

Edwin