Question

Without expansion, prove that the following determinants vanish

$open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space straight a space end cell bc row cell begin inline style 1 over straight b end style end cell cell space straight b end cell cell space ca end cell row cell begin inline style fraction numerator 1 over denominator space straight c end fraction end style end cell cell space straight c end cell cell space ab end cell end table close vertical bar$

Solution

Let space space increment space space space equals space open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space straight a space end cell bc row cell begin inline style 1 over straight b end style end cell cell space straight b end cell cell space ca end cell row cell begin inline style fraction numerator 1 over denominator space straight c end fraction end style end cell cell space straight c end cell cell space ab end cell end table close vertical bar space equals space abc space space space open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space straight a space end cell bc row cell begin inline style 1 over straight b end style end cell cell space straight b end cell cell space ca end cell row cell begin inline style fraction numerator 1 over denominator space straight c end fraction end style end cell cell space straight c end cell cell space ab end cell end table close vertical bar space comma by space dividing space straight C subscript 3 space by space abc space space 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 space space space space space space space space space space space space open square brackets table row cell therefore space two space column space are space identical end cell end table close square brackets Note space colon space If space 1 comma space straight omega comma space straight omega 2 space are space cube space roots space of space unity comma space then space 1 space plus space straight omega space plus space straight omega 2 space equals space 0 comma space straight omega 3 space equals space 1.

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