The system of equations αx + y + z = α − 1 x + α y + z = α − 1 x + y + α z = α − 1 has no solution, if α is

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The system of equations ​ αx + y + z = α − 1 x + α y + z = α − 1 x + y + α z = α − 1 ​ has no solution, if α is

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​ αx + y + z = α − 1 x + α y + z = α − 1 x + y + z α = α − 1 Δ = ​ α 1 1 ​ 1 α 1 ​ 1 1 α ​ ​ = α ( α 2 − 1 ) − 1 ( α − 1 ) + 1 ( 1 − α ) = α ( α − 1 ) ( α + 1 ) − 1 ( α − 1 ) − 1 ( α − 1 ) ⇒ ( α − 1 ) [ α 2 + α − 1 − 1 ] = 0 ⇒ ( α − 1 ) [ α 2 + α − 2 ] = 0 [ α 2 + 2 α − α − 2 ] = 0 ( α − 1 ) [ α ( α + 2 ) − 1 ( α + 2 )] = 0 ( α − 1 ) = 0 , α + 2 = 0 ⇒ α = − 2 , 1 ; but α  = 1 ​

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