1.

Using properties ofdeterminants, prove the following:`|(1,x,x^2),(x^2, 1,x),(x,x^2, 1)|=(1-x^3)^2`

Answer» Let the value of the given determinant be `Delta`. Then,
`Delta = |{:(1, x, x^(2)), (x^(2), 1, x),(x, x^(2), 1):}|`
`= |{:(1+x+x^(2), x, x^(2)), (1+x+x^(2), 1, x),(1+x+x^(2), x^(2), 1):}| ["applying"C_(1) to (C_(1) + C_(2) +C_(3))]`
`=(1+x+x^(2)) |{:(1, x, x^(2)), (1, 1, x),(1, x^(2), 1):}| ["taking"(1+x+x^(2))"common from"C_(1)]`
`=(1+x+x^(2)) |{:(1, " "x, " "x^(2)), (0, 1-x, x(1-x)),(0, x(x-1), (1-x)(1+x)):}| ["applying"R_(2) to (R_(2)-R_(1)) "and"R_(3) to (R_(3)-R_(1))]`
`=(1+x+x^(2))(1-x)^(2)* |{:(1, x, x^(2)), (0, 1, x),(0, -x, (1+x)):}| ["taking(1-x) common from each of "R_(2)" and "R_(3)]`
` =(1+x+x^(2))(1-x)^(2)*1* |{:(1, x), (-x, (1+x)):}| ["expanded by"C_(1)]`
`=(1+x+x^(2))(1-x)^(2) * (1+x+x^(2))`
`={(1-x)(1+x+x^(2))}^(2) = (1-x^(3))^(2)`
Hence, `Delta = (1-x^(3))^(2)`


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