Using properties of determinants, prove that where α, β, γ, are in A.P. - Wired Faculty

Question

Using properties of determinants, prove that

$space open square brackets table row cell straight x minus 3 end cell cell space space straight x minus 4 space end cell cell space space straight x minus straight alpha end cell row cell straight x minus 2 end cell cell space space straight x minus 3 end cell cell space space straight x minus straight beta end cell row cell straight x minus 1 end cell cell space space straight x minus 2 end cell cell space space straight x minus straight gamma end cell end table close square brackets space equals 0$

where α, β, γ, are in A.P.

Solution

space Let space increment space equals space open square brackets table row cell straight x minus 3 end cell cell space space straight x minus 4 space end cell cell space space straight x minus straight alpha end cell row cell straight x minus 2 end cell cell space space straight x minus 3 end cell cell space space straight x minus straight beta end cell row cell straight x minus 1 end cell cell space space straight x minus 2 end cell cell space space straight x minus straight gamma end cell end table close square brackets space space space space space space space space space space space space space equals space open square brackets table row cell straight x minus 3 end cell cell space space straight x minus 4 space end cell cell space space straight x minus straight alpha end cell row 0 cell space space 0 end cell cell space space minus 2 straight beta plus straight a plus straight gamma end cell row cell straight x minus 1 end cell cell space space straight x minus 2 end cell cell space space straight x minus straight gamma end cell end table close square brackets space comma space by space straight R subscript 2 space rightwards arrow space 2 space space straight R subscript 2 space minus straight R subscript 1 space minus straight R subscript 1 space space space space space space space space space space equals space space open square brackets table row cell straight x minus 3 end cell cell space space straight x minus 4 space end cell cell space space straight x minus straight alpha end cell row 0 0 0 row cell straight x minus 1 end cell cell space space straight x minus 2 end cell cell space space straight x minus straight gamma end cell end table close square brackets space space space space space space space space open square brackets table row cell therefore space 2 beta equals alpha plus gamma space as space straight alpha comma straight beta comma straight gamma space are space in space straight A. straight P space end cell end table close square brackets space space 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 open square brackets table row cell therefore space each space element space of space second space row space is space zero end cell end table close square brackets

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