Without expanding, prove that the following determinants vanish :

$open vertical bar table row 5 cell space space 15 end cell cell space space minus 25 end cell row cell 7 space space end cell cell space space 21 end cell cell space space space space space space 30 end cell row 8 cell space space 24 end cell cell space space space space space 42 end cell end table close vertical bar$

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Solution

Let space increment space equals space space open vertical bar table row 5 cell space space 15 end cell cell space space minus 25 end cell row cell 7 space space end cell cell space space 21 end cell cell space space space space space space 30 end cell row 8 cell space space 24 end cell cell space space space space space 42 end cell end table close vertical bar space equals space open vertical bar table row 5 cell space space space space space 0 end cell cell space space space minus 25 end cell row 7 cell space space space space space 0 end cell cell space space space 30 end cell row 8 cell space space space space space 0 end cell cell space space space space 42 end cell end table close vertical bar space comma space by space straight C subscript 2 space 3 straight C subscript 1 space space space space space space space space space space equals space 0 space space space space space open square brackets table row cell because space each space element space of space 2 nd space column space is space zero end cell end table close square brackets

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