Question

Prove that:
$open vertical bar table row 1 cell space space straight x end cell cell space space yz end cell row 1 cell space space straight y end cell cell space space zx end cell row 1 cell space straight z end cell cell space space xy end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis$

Solution

Let $increment space equals space open vertical bar table row 1 cell space space straight x end cell cell space yz end cell row 1 cell space straight y end cell cell space zx end cell row 1 cell space straight z end cell cell space xy end cell end table close vertical bar space equals space 1 over xyz space open vertical bar table row straight x cell space space straight x squared end cell cell space space xyz end cell row straight y cell space straight y squared end cell cell space space xyz end cell row straight z cell space straight z squared end cell cell space xyz end cell end table close vertical bar$ ,
by multiplying $straight R subscript 1 comma space straight R subscript 2 comma space straight R subscript 3 space and space with space straight a comma space straight b comma space straight c space respectively.$
$equals xyz over xyz open vertical bar table row straight x cell space space straight x squared end cell cell space 1 end cell row straight y cell space straight y squared end cell cell space 1 end cell row straight z cell space straight z squared end cell cell space 1 end cell end table close vertical bar equals space open vertical bar table row 1 cell space space straight x end cell cell space space straight x squared end cell row 1 cell space space straight y end cell cell space space straight y squared end cell row 1 cell space space straight z end cell cell space space straight z squared end cell end table close vertical bar comma space space by space passing space straight R subscript 3 space over space two space rows. equals space open vertical bar table row 1 cell space space space straight x end cell cell space space space space space space straight x squared end cell row 0 cell space space space straight y minus straight x end cell cell space space space space space space straight y squared minus straight x squared end cell row 0 cell space space straight z minus straight x end cell cell space space space space space straight z squared minus straight x squared end cell end table close vertical bar comma space space by space straight R subscript 2 minus straight R subscript 1 comma space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight y minus straight x end cell cell space space space space space space space straight y squared minus straight x squared end cell row cell straight z minus straight x end cell cell space space space space space space straight z squared minus straight x squared end cell end table close vertical bar comma space by space expanding space with space straight C subscript 1 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 1 cell space space space straight y plus straight x end cell row 1 cell space space space straight z plus straight x 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 plus straight x minus straight y minus straight x right parenthesis 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 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.$

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