Factorise the determinant:
$open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight a cell space space straight b end cell cell space straight c end cell row cell straight a squared end cell cell space space straight b squared end cell cell space space straight c squared end cell end table close vertical bar.$

Solution

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