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Let P = [{:( - 30, 20, 56), ( 90, 140, 1...

Let `P = [{:( - 30, 20, 56), ( 90, 140, 112), ( 120, 60, 14):}] and A = [{:( 2,7,omega ^(2)), ( -1, - omega , 1), ( 0, - omega, - omega + 1):}] where omega = ( -1 + sqrt3)/(2), and I _(3)` be the identity matrix of order 3. If the determinate of the matrix `(P ^(-1) AP - I_(3)) ^(2) ` is `a omega ^(2),` then the value of `alpha ` is equal to` "_________."`

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