Without expanding, prove that

$open vertical bar table row 9 cell space space space space space space 9 end cell cell space space 12 end cell row 1 cell space space minus 3 end cell cell space space minus 4 end cell row 1 cell space space space space space space 9 end cell cell space 12 end cell end table close vertical bar space equals 0$

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Solution

Let space equals space open vertical bar table row 9 cell space space space space space space 9 end cell cell space space 12 end cell row 1 cell space space minus 3 end cell cell space space minus 4 end cell row 1 cell space space space space space space 9 end cell cell space 12 end cell end table close vertical bar space equals begin inline style 1 third end style space open vertical bar table row 9 cell space space space space space space 9 end cell cell space space 12 end cell row 1 cell space space minus 3 end cell cell space space minus 4 end cell row 1 cell space space space space space space 9 end cell cell space 12 end cell end table close vertical bar space comma space multiply space straight C subscript 3 space end subscript by space 3 space space space space space space space equals begin inline style 1 third end style space open vertical bar table row 9 cell space space space space space space 9 end cell cell space space 12 end cell row 1 cell space space minus 3 end cell cell space space minus 4 end cell row 1 cell space space space space space space 9 end cell cell space 12 end cell end table close vertical bar space comma space by space straight C subscript 3 space end subscript minus 4 space straight C subscript 2 space space space space space space space space equals 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 open square brackets table row cell because space each space element space of space third space column space is space zero end cell end table close square brackets

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