Question

Without expanding, prove that the following determinants vanish :

$open square brackets table row 102 cell space 18 end cell cell space space 36 end cell row 1 cell space 3 end cell cell space 4 end cell row 17 cell space 3 end cell cell space 6 end cell end table close square brackets$

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Solution

Let space space space increment space equals space open square brackets table row 102 cell space 18 end cell cell space space 36 end cell row 1 cell space 3 end cell cell space 4 end cell row 17 cell space 3 end cell cell space 6 end cell end table close square brackets space equals space open vertical bar table row cell 6 cross times 17 end cell cell space space 6 cross times 3 space space end cell cell 6 cross times 6 end cell row 1 3 4 row 17 3 6 end table close vertical bar space equals space 6 space open vertical bar table row 17 cell space space space 3 end cell cell space space 6 end cell row 1 cell space space space 3 end cell cell space space 4 end cell row 17 cell space space space 3 end cell cell space space 6 end cell end table close vertical bar space space space space space space space space space space space space space equals space 6 space cross times 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 space space space space space space space space space space open square brackets table row cell because space straight R subscript 1. space straight R subscript 3 end cell end table space are space identical close square brackets space space space space space space space space space space space space space equals 0 space space space space

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