For a real number α, if the system ⎡⎢⎣1 α α² α 1 α α² α 1⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦ = ⎡⎢⎣1 k⎤⎥⎦ of linear equations, has infinitely many solutions, then 1+α+α²=

Given system of linear equations has infinitely many solutions
∴ Δ=0
⇒ ∣∣∣1 α α²α 1 α²α 1∣∣∣ = 0
⇒ 1(1−α⁴)−α(α−α³)+α²(α²−α²)=0
⇒ 1−α⁴−α²+α⁴=0
⇒ 1−α²=0
⇒ α²=1
⇒ α=±1
If α=1, then the matrix reduces to
⎡⎢⎣1 1 11 1 11 1⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣1 k⎤⎥⎦
which is not possible since x+y+z obtains two different values
∴ α=1 is not possible
⇒ α=−1
∴ 1+α+α²=1−1+1=1
Hence, 1+α+α²=1