Without expanding, prove that the following determinants vanish :

$open square brackets table row 6 cell space space 0 space end cell cell space space 2 end cell row 9 cell space 1 end cell cell space space 3 end cell row 15 cell space 2 end cell cell space space 5 end cell end table close square brackets$

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Solution

Let space space increment space equals space open square brackets table row 6 cell space space 0 space end cell cell space space 2 end cell row 9 cell space 1 end cell cell space space 3 end cell row 15 cell space 2 end cell cell space space 5 end cell end table close square brackets space equals space open square brackets table row 0 cell space space space 0 end cell cell space space 2 end cell row 0 cell space space 1 end cell cell space space 3 end cell row 0 cell space space 2 end cell cell space space 5 end cell end table close square brackets space comma space by space straight C subscript 1 space end subscript minus space straight C subscript 3 space space space space space space space space space space space space equals 0 space space space space space open square brackets table row because end table space each space element space of space first space column space is space zero close square brackets

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