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Prove that : |{:((y+z)^(2),x^(2),x^(2)),...

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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`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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Knowledge Check

  • (x+y)^(2)-z^(2)=-9, (y+z)^(2)-x^(2)=15, (z+x)^(2)-y^(2)=3

    A
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    B
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    C
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    D
    none

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