Question

Without expanding, prove that the following determinants vanish :

$open vertical bar table row 1 cell space space straight a end cell cell space space straight b plus straight c end cell row 1 cell space space straight b end cell cell space space straight c plus straight a end cell row 1 cell space space straight c end cell cell space space straight a plus straight b end cell end table close vertical bar$

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Solution

Let space space increment space equals space space open vertical bar table row 1 cell space space straight a end cell cell space space straight b plus straight c end cell row 1 cell space space straight b end cell cell space space straight c plus straight a end cell row 1 cell space space straight c end cell cell space space straight a plus straight b end cell end table close vertical bar space equals space space open vertical bar table row 1 cell space space straight a plus straight b plus straight c end cell cell space space straight b plus straight c end cell row 1 cell space space straight a plus straight b plus straight c end cell cell space space straight c plus straight a end cell row 1 cell space space straight a plus straight b plus straight c end cell cell space space straight a plus straight b end cell end table close vertical bar space comma by space straight C subscript 2 space plus straight C subscript 3 Taking space out space common space straight a space plus space straight b space plus space straight c space from space straight C subscript 2 equals left parenthesis straight a plus straight b plus straight c right parenthesis space cross times space 0 space space space space space space space space space space space space space space space space space space space space space open square brackets table row cell because space straight C subscript 1 comma straight C subscript 2 space are space identical end cell end table close square brackets equals space 0

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