Question

Using the properties of determinants, prove that
$open vertical bar table row cell straight a plus straight b plus straight c end cell cell negative straight c end cell cell negative straight b end cell row cell negative straight c end cell cell space space space straight a plus straight b plus straight c end cell cell negative straight a end cell row cell negative straight b end cell cell negative straight a end cell cell space space straight a plus straight b plus straight c end cell end table close vertical bar space equals 2 left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis$

Solution

Let space space increment space equals space open vertical bar table row cell straight a plus straight b plus straight c end cell cell negative straight c end cell cell negative straight b end cell row cell negative straight c end cell cell space space space straight a plus straight b plus straight c end cell cell negative straight a end cell row cell negative straight b end cell cell negative straight a end cell cell space space straight a plus straight b plus straight c end cell end table close vertical bar space space space space space space space space space space space space space equals space space open vertical bar table row cell straight a plus straight b plus straight c end cell cell negative straight c end cell cell negative straight b end cell row cell negative straight c end cell cell space space space straight a plus straight b plus straight c end cell cell negative straight a end cell row cell negative straight b end cell cell negative straight a end cell cell space space straight a plus straight b plus straight c end cell end table close vertical bar space space comma space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 space space space space space space space space space space space space equals space left parenthesis straight a plus straight b right parenthesis left parenthesis straight b plus straight c right parenthesis space space open vertical bar table row 1 cell negative 1 end cell cell negative straight b end cell row 1 cell space space space 1 end cell cell negative straight a end cell row cell negative 1 end cell cell space space 1 end cell cell space space straight a plus straight b plus straight c end cell end table close vertical bar space space space space space space space space space space space space equals space left parenthesis straight a plus straight b right parenthesis left parenthesis straight b plus straight c right parenthesis space open vertical bar table row 1 cell negative 1 end cell cell negative straight b end cell row 0 cell space space space 2 end cell cell space minus straight a plus straight b end cell row 0 cell space space 0 end cell cell space space space space straight c plus straight a end cell end table close vertical bar space comma space by space straight R subscript 2 minus straight R subscript 1 comma space straight R subscript 3 space plus straight R subscript 1 space space space space space space space space space space space space equals space left parenthesis straight a plus straight b right parenthesis left parenthesis straight a plus straight b right parenthesis open square brackets table row cell left parenthesis 1 right parenthesis left parenthesis 2 right parenthesis left parenthesis straight c plus straight a right parenthesis end cell end table close square brackets space space space space space space space space space space space space equals space 2 space space left parenthesis straight a plus straight b right parenthesis space left parenthesis straight b plus straight c right parenthesis left parenthesis straight c plus straight a right parenthesis

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