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for a real number alpha if the system ...

for a real number `alpha` if the system
`[{:(1,,alpha,,alpha^(2)),(alpha,,1,,alpha),(alpha^(2),,alpha,,1):}][{:(x),(y),(z):}]=[{:(1),(-1),(1):}]` of linear equations has infinitely many solutions then `1+alpha+alpha^(2) = "______"`

Text Solution

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The correct Answer is:
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`
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