InterviewSolution
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InterviewSolution
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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)` |
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Answer» `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]` `=-2(x^3+y^3+z^3-3xyz) = R.H.S.` |
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