Question

Without expansion, prove that the following determinants vanish
$open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space space space space straight a squared end cell cell space space bc end cell row cell begin inline style 1 over straight b end style end cell cell space space space straight b squared end cell cell space space ca end cell row cell begin inline style 1 over straight c end style end cell cell space space straight c squared end cell cell space space ab end cell end table close vertical bar$

Solution

Let space space space increment space equals space open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space space space space straight a squared end cell cell space space bc end cell row cell begin inline style 1 over straight b end style end cell cell space space space straight b squared end cell cell space space ca end cell row cell begin inline style 1 over straight c end style end cell cell space space straight c squared end cell cell space space ab end cell end table close vertical bar space equals abc space space space open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space space space space straight a squared end cell cell begin inline style 1 over straight a end style end cell row cell begin inline style 1 over straight b end style end cell cell space space space straight b squared end cell cell begin inline style 1 over straight b end style end cell row cell begin inline style 1 over straight c end style end cell cell space space straight c squared end cell cell begin inline style 1 over straight c end style end cell end table close vertical bar space comma space by space diving space straight C subscript 3 space by space abc space space space space space space space space space space space equals 0 space space space space space space space space space space space space space space open square brackets table row cell because space two space column space are space identical end cell end table close square brackets

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