Using properties of determinants, prove that
$open vertical bar table row straight x cell space straight x squared end cell cell space yz end cell row straight y cell space straight y squared end cell cell space zx end cell row straight z cell space straight z squared end cell cell space xy end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis xy plus yz plus zx right parenthesis$

Solution

L.H.S. = $open vertical bar table row straight x cell space straight x squared end cell cell space yz end cell row straight y cell space straight y squared end cell cell space zx end cell row straight z cell space straight z squared end cell cell space xy end cell end table close vertical bar$
$equals space 1 over xyz open vertical bar table row cell straight x squared end cell cell space space straight x cubed end cell cell space space xyz end cell row cell straight y squared end cell cell space space straight y cubed end cell cell space space xyz end cell row cell straight z squared end cell cell space space straight z cubed end cell cell space xyz end cell end table close vertical bar space equals space xyz over xyz space open vertical bar table row cell straight x squared end cell cell space straight x cubed end cell cell space space 1 end cell row cell straight y squared end cell cell space straight y cubed end cell cell space 1 end cell row cell straight z squared end cell cell space straight z cubed end cell cell space 1 end cell end table close vertical bar space equals space open vertical bar table row 1 cell space straight x squared end cell cell space straight x cubed end cell row 1 cell space straight y squared end cell cell space straight y cubed end cell row 1 cell space straight z squared end cell cell space straight z cubed end cell end table close vertical bar equals space open vertical bar table row 1 cell space space space straight x squared end cell cell space space space straight x cubed end cell row 0 cell space space space space straight y squared minus straight x squared end cell cell space space space space space straight y cubed minus straight x cubed end cell row 0 cell space space space straight z squared minus straight x squared end cell cell space space straight z cubed minus straight x cubed end cell end table close vertical bar comma space by space straight R subscript 2 minus straight R subscript 2 comma space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight y squared minus straight x squared end cell cell space space space space straight y cubed minus straight x cubed end cell row cell straight z squared minus straight x squared end cell cell space space space space straight z cubed minus straight x cubed end cell end table close vertical bar space equals space space open vertical bar table row cell left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight y plus straight x right parenthesis end cell cell left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight y squared plus straight x squared plus yx right parenthesis end cell row cell left parenthesis straight z minus straight x right parenthesis space left parenthesis straight z plus straight x right parenthesis end cell cell left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z squared plus straight x squared plus zx right parenthesis end cell end table close vertical bar equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis space open vertical bar table row cell straight y plus straight x end cell cell space space space space space space space space straight y squared plus straight x squared plus yx end cell row cell straight z plus straight x end cell cell space space space space space space space straight z squared plus straight x squared plus zx end cell end table close vertical bar equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space open vertical bar table row cell straight y plus straight x end cell cell space straight y squared plus straight x squared plus yx end cell row cell straight z minus straight y end cell cell space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight z plus straight y right parenthesis space plus straight x space left parenthesis straight z minus straight y right parenthesis end cell end table close vertical bar space equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis space open vertical bar table row cell straight y plus straight x end cell cell space space space space space space space space straight y squared plus straight x squared plus yx end cell row 1 cell space space space space space space straight z plus straight y plus straight x end cell end table close vertical bar equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space open square brackets yz plus straight y squared plus xy plus zx plus xy plus straight x squared minus straight y squared minus straight x squared minus xy close square brackets equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis xy plus yz plus zx right parenthesis equals space straight R. straight H. straight S.$

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