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Determinants

Question
CBSEENMA12034364
Wired Faculty App

 Without expanding, prove that the following determinants vanish :
 
open vertical bar table row cell straight a minus straight b end cell cell space space straight b minus straight c space end cell cell straight c minus straight a end cell row cell straight b minus straight c end cell cell space straight c minus straight a end cell cell space straight a minus straight b end cell row cell straight c minus straight a end cell cell space straight a minus straight b end cell cell space straight b minus straight c end cell end table close vertical bar

Solution
Let space increment space equals space open vertical bar table row cell straight a minus straight b end cell cell space space straight b minus straight c space end cell cell straight c minus straight a end cell row cell straight b minus straight c end cell cell space straight c minus straight a end cell cell space straight a minus straight b end cell row cell straight c minus straight a end cell cell space straight a minus straight b end cell cell space straight b minus straight c 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 0 cell space 0 end cell 0 row cell straight b minus straight c end cell cell space straight c minus straight a end cell cell space straight a minus straight b end cell row cell straight c minus straight a end cell cell space straight a minus straight b end cell cell space straight b minus straight c end cell end table close vertical bar space space space space comma space by space straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3 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 space space space space space space space open square brackets table row cell because space each space element space of space first space row space is space zero end cell end table close square brackets space

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