Question

Without expanding, prove that the following determinants vanish :

$open vertical bar table row 43 cell space space 1 end cell cell space space space 6 end cell row 28 cell space space 7 end cell cell space space space 4 end cell row 14 cell space space 3 end cell cell space space space 2 end cell end table close vertical bar$

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Solution

Let space increment space equals space space space open vertical bar table row 43 cell space space 1 end cell cell space space space 6 end cell row 28 cell space space 7 end cell cell space space space 4 end cell row 14 cell space space 3 end cell cell space space space 2 end cell end table close vertical bar space equals space open square brackets table row cell 7 cross times 6 end cell cell space space space 1 end cell cell space space space space 6 end cell row cell 7 cross times 4 end cell cell space space space 7 end cell cell space space space 4 end cell row cell 7 cross times 2 end cell cell space space space 3 end cell cell space space space 2 end cell end table close square brackets space equals space 7 space open square brackets table row 6 cell space space 1 end cell cell space space 6 end cell row 4 cell space space 7 end cell cell space space 4 end cell row 2 cell space space 3 end cell cell space space 2 end cell end table close square brackets space equals space 0 space open square brackets table row cell because space two space columns space are space identical end cell end table close square brackets 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

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