Question

Without expanding, prove that the following determinants vanish :

$open vertical bar table row straight x cell space space space straight a end cell cell space space space straight x plus straight a end cell row straight y cell space space space straight b end cell cell space space straight y plus straight b end cell row straight z cell space space space straight c end cell cell space space straight z plus straight c end cell end table close vertical bar space equals 0$

Solution

straight L. straight H. straight S space equals space open vertical bar table row straight x cell space space space straight a end cell cell space space space straight x plus straight a end cell row straight y cell space space space straight b end cell cell space space straight y plus straight b end cell row straight z cell space space space straight c end cell cell space space straight z plus straight c end cell end table close vertical bar space equals space space open vertical bar table row cell straight x space plus straight a end cell cell space space space straight a end cell cell space space space straight x plus straight a end cell row cell straight y plus straight b end cell cell space space space straight b end cell cell space space straight y plus straight b end cell row cell straight z plus straight c end cell cell space space space straight c end cell cell space space straight z plus straight c end cell end table close vertical bar space by space straight C subscript 1 rightwards arrow straight C subscript 1 space plus straight C subscript 2 space space space space space space space space space space space equals 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 space space space space space space space space space space space open square brackets table row cell because space straight C subscript 1 end cell end table comma space straight C subscript 3 space are space identical close square brackets space space space space space space space space space space space space space space space space equals space straight R. straight H. straight S

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