Question

$Without space expanding comma space prove space that space the space following space determinants space vanish space open vertical bar table row 1 cell space space space space space space bc end cell cell space space space space space straight a left parenthesis straight b plus straight c right parenthesis end cell row 1 cell space space space space space ca end cell cell space space space space space straight b left parenthesis straight c plus straight a right parenthesis end cell row 1 cell space space space space space space ab end cell cell space space space space space straight c left parenthesis straight a plus straight b right parenthesis end cell end table close vertical bar$

Solution

Let space increment space equals open vertical bar table row 1 cell space space space space space space bc end cell cell space space space space space straight a left parenthesis straight b plus straight c right parenthesis end cell row 1 cell space space space space space ca end cell cell space space space space space straight b left parenthesis straight c plus straight a right parenthesis end cell row 1 cell space space space space space space ab end cell cell space space space space space straight c left parenthesis straight a plus straight b right parenthesis end cell end table close vertical bar space space space space space space space space space space space equals space open vertical bar table row 1 cell space space space space space space bc end cell cell space space space space ab plus bc plus ca end cell row 1 cell space space space space space ca end cell cell space space space ab plus bc plus ca end cell row 1 cell space space space space space space ab end cell cell space space space ab plus bc plus ca end cell end table close vertical bar space comma space by space straight C subscript 3 space end subscript plus straight C subscript 2 space space space space space space space space space space space equals space left parenthesis straight a plus bc plus ca right parenthesis space space open vertical bar table row 1 cell space space space space space space bc end cell cell space space space space 1 end cell row 1 cell space space space space space ca end cell cell space space space space 1 end cell row 1 cell space space space space space space ab end cell cell space space space space 1 end cell end table close vertical bar space space space space space space space space space space space space equals left parenthesis ab plus bc plus ca right parenthesis space cross times space 0 space space space space space space space space space space space space equals space 0 space space space space space space space space space space space space open square brackets table row cell because space two space columns space are space identical end cell end table close square brackets

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