Question

For a real number `alpha,` if the system `[1alphaalpha^2alpha1alphaalpha^2alpha1][x y z]=[1-1 1]` of linear equations, has infinitely many solutions, then `1+alpha+alpha^2=`

Answer 1

Correct Answer - 1
`|{:(1, alpha, alpha^(2)), (alpha, 1, alpha), (alpha^(2), alpha, 1):}| =0`
`rArr alpha^(4)-2alpha^(2) +1 =0`
`rArr alpha^(2) = 1`
`rArr alpha = +-1`
But `alpha = 1` not possible [Not satisfying equation]
`therefore alpha = -1`
Hence, `1+alpha + alpha^(2) +1`