Question

Without expanding, prove that the following determinants vanish :

$open vertical bar table row 43 cell space space space 1 end cell cell space space space space 6 end cell row 35 cell space space space 7 end cell cell space space space space 4 end cell row 17 cell space space space 3 end cell cell space space space 2 end cell end table close vertical bar space$

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Solution

Let space space increment space space equals space open vertical bar table row 43 cell space space space 1 end cell cell space space space space 6 end cell row 35 cell space space space 7 end cell cell space space space space 4 end cell row 17 cell space space space 3 end cell cell space space space 2 end cell end table close vertical bar space equals space open vertical bar table row 42 cell space space 1 end cell cell space space space 6 end cell row 28 cell space space space 7 end cell cell space space space 4 end cell row 14 cell space space space 3 space end cell cell space space space 2 end cell end table close vertical bar space comma space by space straight C subscript 1 space minus straight C subscript 2 space space space space space space space space space space space space equals space open vertical bar table row 0 cell space space 1 end cell cell space space 6 end cell row 0 cell space space 7 end cell cell space space 4 end cell row 0 cell space 3 end cell cell space space 2 end cell end table close vertical bar space comma space by space straight C subscript 1 space minus 7 straight C subscript 3 space space space space space space space space space space space equals space 0 space space space space space space open square brackets table row cell because space each space element space of space first space column space is space zero end cell end table close square brackets

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