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Prove |(y+z,z+x,x+y),(z+x,x+y,y+z),(x+y,...

Prove `|(y+z,z+x,x+y),(z+x,x+y,y+z),(x+y,y+z,z+x)|=2|(x,y,z),(y,z,x),(z,x,y)|=-2(x^3+y^3+z^3-3xyz)`

Text Solution

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`L.H.S. = |[y+z,z+x,x+y],[z+x,x+y,y+z],[x+y,y+z,z+x]|`
`=|[y,z,x],[z,x,y],[x,y,z]|+|[z,x,y],[x,y,z],[y,z,x]|`
`=(-1)|[y,x,z],[z,y,x],[x,z,y]|+(-1)|[x,z,y],[y,x,z],[z,y,x]|`
`=(-1)^2|[x,y,z],[y,z,x],[z,x,y]|+(-1)^2|[x,y,z],[y,z,x],[z,x,y]|`
`=|[x,y,z],[y,z,x],[z,x,y]|+|[x,y,z],[y,z,x],[z,x,y]|`
`=2|[x,y,z],[y,z,x],[z,x,y]|` ...(first part proved)
`=2[x(zy-x^2)-y(y^2-zx)+z(yx-z^2)]`
`=2[3xyz-x^3-y^3-z^3]`
...
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