By using properties of determinants, show that:
$open vertical bar table row 1 cell space space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space straight b end cell cell space space space straight b squared end cell row 1 cell space space straight c end cell cell space space space straight c squared end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis space left parenthesis straight c minus straight a right parenthesis$

Solution

Let $increment space equals space open vertical bar table row 1 cell space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space straight b end cell cell space space space straight b squared end cell row 1 cell space space space straight c end cell cell space space space straight c squared end cell end table close vertical bar$
$equals space open vertical bar table row 1 cell space space space straight a end cell cell space space space straight a squared end cell row 0 cell space space straight b minus straight a end cell cell space space space straight b squared minus straight a squared end cell row 0 cell space space space straight c minus straight a end cell cell space space space space straight c squared minus straight a squared end cell end table close vertical bar comma space space by space straight R subscript 2 space minus space straight R subscript 1 comma space space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight b minus straight a end cell cell space space space space space space straight b squared minus straight a squared end cell row cell straight c minus straight a end cell cell space space space space space space straight c squared minus straight a squared end cell end table close vertical bar comma space space expanding space by space first space column space equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space open vertical bar table row 1 cell space space space straight b plus straight a end cell row 1 cell space space space space straight c plus straight a end cell end table close vertical bar comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space by space taking space straight b minus straight a space common space from space straight R subscript 1 space and space straight c minus straight a space from space space straight R subscript 2. equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space left square bracket straight c plus straight a minus straight b minus straight a right square bracket equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c minus straight b right parenthesis space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis.$

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