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Prove that : `|{:((y+z)^(2),x^(2),x^(2)),(y^(2),(x+z)^(2),y^(2)),(z^(2),z^(2),(x+y)^(2)):}|=2xyz (x+y+z)^(3)` |
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Answer» `LHS=|{:((y+z)^(2),x^(2),y^(2)),(y^(2), (z+x)^(2),y^(2)),(z^(2) ,z^(2) ,(x+y)^(2)):}|` Apply `C_(2) to C_(1), C_(3) to C_(3) - C_(1)` `|{:(,(y+z)^(2),x^(2) - (y+z)^(2) ,x^(2)-(y+z)^(2)),(,y^(2),(z+x)^(2)-y^(2) , 0),(,z^(2) ,0,(x+y)^(2)-z^(2)):}|` `|{:(,(y+z^(2)),(x+y+z),(x+y+z)(x-y-z)),(,y^(2) ,(z+x+y)(z+x-y),0),(,z^(2) ,0,(x+y+z)(x+y-z)):}|` Taking (x+x+y) common form `C_(2)` as well as `c_(3)` `=(x+y +z)^(2)|{:((y+z)^(2),(x-y-z),(x-y-z)),(y^(2),(z+x-y),0),(z^(2),0,(x+y+z)):}|` Apply `R_1 to R_2 to R_3` `= (x+y+z)^(2) |{:(2yz,,-2z,,-2y),(y^(2),,(z+x+y),,0),(z^(2),,0,,(x+y-z)):}|` |
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