Using properties of determinants, prove that
$open vertical bar table row cell straight b plus straight c end cell cell space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight c plus straight a end cell cell space straight a plus straight b end cell cell space straight b plus straight c end cell row cell straight a plus straight b end cell cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar space equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis ab plus bc plus ca minus straight a squared minus straight b squared minus straight c squared right parenthesis$

Solution

L.H.S. = $open vertical bar table row cell straight b plus straight c end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell straight c plus straight a end cell cell straight a plus straight b end cell cell straight b plus straight c end cell row cell straight a plus straight b end cell cell straight b plus straight c end cell cell straight c plus straight a end cell end table close vertical bar space equals space open vertical bar table row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight a plus straight b end cell cell straight b plus straight c end cell row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight b plus straight c end cell cell straight c plus straight a end cell end table close vertical bar comma space space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3$
$equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 cell straight c plus straight a end cell cell space straight a plus straight b end cell row 1 cell straight a plus straight b end cell cell space straight b plus straight c end cell row 1 cell straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar space equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space open vertical bar table row 1 cell space straight c plus straight a end cell cell space straight a plus straight b end cell row 0 cell space straight b minus straight c end cell cell space straight c minus straight a end cell row 0 cell space straight b minus straight a end cell cell space straight c minus straight b end cell end table close vertical bar comma space space space by space straight R subscript 2 space minus straight R subscript 1 comma space space straight R subscript 3 minus straight R subscript 1 equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row cell straight b minus straight c end cell cell space space straight c minus straight a end cell row cell straight b minus straight a end cell cell space space straight c minus straight b end cell end table close vertical bar equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space left square bracket left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight b right parenthesis space minus left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis right square bracket equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space left square bracket negative straight b squared minus straight c squared plus 2 bc minus bc plus ab plus ca minus straight a squared right square bracket equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis space left parenthesis ab plus bc plus ca minus straight a squared minus straight b squared minus straight c squared right parenthesis equals space straight R. straight H. straight S.$

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