Without expanding, prove that the following determinants vanish :
$open vertical bar table row cell straight x plus straight y end cell cell space space space straight y plus straight z end cell cell space space space straight z plus straight x end cell row straight z straight x straight y row 1 1 1 end table close vertical bar$

increment space equals space open vertical bar table row cell straight x plus straight y end cell cell space space space straight y plus straight z end cell cell space space space straight z plus straight x end cell row straight z straight x straight y row 1 1 1 end table close vertical bar space space space space space equals space open vertical bar table row cell straight x plus straight y end cell cell space space space straight y plus straight z end cell cell space space space straight z plus straight x end cell row straight z straight x straight y row 1 1 1 end table close vertical bar space comma space by space straight R subscript 1 space rightwards arrow space straight R subscript 1 space plus straight R subscript 2 space space space equals 0 space space open square brackets table row cell because space element space of space space space straight R subscript 1 comma straight R subscript 3 end cell end table space are space portional close square brackets

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Determinants

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