Using the properties of determinants, show that:
$open vertical bar table row 1 cell space space space straight x plus straight y end cell cell space space space straight x squared plus straight y squared end cell row 1 cell space straight y plus straight z end cell cell space space straight y squared plus straight z squared end cell row 1 cell space straight z plus straight x end cell cell space space straight z squared plus straight x squared end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis$

Solution

Let space space increment space equals space open vertical bar table row 1 cell space space space straight x plus straight y end cell cell space space space straight x squared plus straight y squared end cell row 1 cell space space straight y plus straight z end cell cell space space space straight y squared plus straight z squared end cell row 1 cell space straight z plus straight x end cell cell space space space straight z squared plus straight x squared end cell end table close vertical bar space space space space space space space space space space

equals space open vertical bar table row 1 cell space space straight x plus straight y end cell cell space space space straight x squared plus straight y squared end cell row 0 cell space space straight z minus straight x end cell cell space space space straight z squared minus straight x squared end cell row 0 cell space space straight z minus straight y end cell cell space space space straight z squared minus straight y squared end cell end table close vertical bar space by space straight R subscript 2 minus straight R subscript 1 comma space space space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight z minus straight x end cell cell space space space space straight z squared minus straight x squared end cell row cell straight z minus straight y end cell cell space space space space straight z squared minus straight y squared end cell end table close vertical bar comma space by space space expanding space with space straight C subscript 1 equals space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis thin space open vertical bar table row 1 cell space space space straight z plus straight x end cell row 1 cell space space space space straight z plus straight y end cell end table close vertical bar space equals space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight z plus straight y minus straight z minus straight x right parenthesis equals space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight y minus straight x right parenthesis 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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