Without expanding, prove that vanishes, where ω, ω 2 are imaginary cube roots of unity. - Wired Faculty

Question

Without expanding, prove that
$open vertical bar table row 1 cell space straight omega end cell cell space space straight omega squared end cell row straight omega cell space space straight omega squared end cell 1 row cell straight omega squared end cell cell space 1 end cell cell space straight omega end cell end table close vertical bar$

vanishes, where ω, ω² are imaginary cube roots of unity.

Solution

Let space increment space equals open vertical bar table row 1 cell space straight omega end cell cell space space straight omega squared end cell row straight omega cell space space straight omega squared end cell 1 row cell straight omega squared end cell cell space 1 end cell cell space straight omega end cell end table close vertical bar space equals space open vertical bar table row cell 1 plus omega plus omega squared end cell omega cell omega squared end cell row cell 1 plus omega plus omega squared end cell cell space space omega squared space space end cell 1 row cell 1 plus omega plus omega squared end cell 1 omega end table close vertical bar space comma space by space straight C subscript 1 plus space straight C subscript 2 space plus straight C subscript 3 space space space space space space space space space space space space equals open vertical bar table row 1 cell space straight omega end cell cell space space straight omega squared end cell row straight omega cell space space straight omega squared end cell 1 row cell straight omega squared end cell cell space 1 end cell cell space straight omega end cell end table close vertical bar space space space space as space 1 plus straight omega plus straight omega squared space equals 0 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 open square brackets table row cell therefore space all space the space elements space of space one space column space are space zero end cell end table close square brackets

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Determinants

Without expanding, prove that vanishes, where ω, ω2 are imaginary cube roots of unity.

Some More Questions From Determinants Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation