Question

Without expanding, prove that the following determinants vanish :

$open vertical bar table row 2 cell space space space 7 end cell cell space space 65 end cell row 3 cell space space space 8 end cell cell space space 75 end cell row 5 cell space space space 9 end cell cell space 86 end cell end table close vertical bar$

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Solution

space straight L. straight H. straight S space equals open vertical bar table row 2 cell space space space 7 end cell cell space space 65 end cell row 3 cell space space space 8 end cell cell space space 75 end cell row 5 cell space space space 9 end cell cell space 86 end cell end table close vertical bar space equals space open vertical bar table row 2 cell space space space 7 end cell cell space space space 0 end cell row 3 cell space space 8 end cell cell space space space 0 end cell row 5 cell space space 9 end cell cell space space space 0 end cell end table close vertical bar space comma space b y space straight C subscript 3 space rightwards arrow space straight C subscript 3 space minus 9 space straight C subscript 2 space minus space straight C subscript 1 space space space 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 open square brackets table row cell because space e a c h space e l e m e n t space o f space straight C subscript 3 end cell end table i s space z e r o close square brackets 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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