1.

Prove that : `=2|{:(1,1,1),(x,y,rz),(x^(2),y^(2),z^(2)):}|=(x-y)(y-z)(z-x)`

Answer» `L.H.S.=2|{:(1,1,1),(x,y,rz),(x^(2),y^(2),z^(2)):}|`
`=|{:(0,0,1),(x-y,y-z,z),(x^(2)-y^(2),y^(2)-z^(2),z^(2)):}|{:(((C_(1)to,C_(1)-,C_(2)))),(((C_(2)to,C_(2)-,C_(3)))):}`
`(x-y)(y-z)|{:(0,0,1),(1,1,z),(x+y,y+z,z^(2)):}|`
`(x-y)(y-z)1|underset(x+y)(1)" "underset(y+z)(1)|`
`("Expending with "R_(1))`
` =(x-y){(y-z)-(x+y))}`
`=(x-y) (y-z) (z-x) = R.H.S.`


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