The value of |α| for which the system of equation αx+y+z=α−1x+αy+z=α−1x+y+αz=α−1has no solution, is _____________

System of equationsαx+y+z=α−1x+αy+z=α−1x+y+αz=α−1Since system has no solution. Therefore, (1) Δ=0 and (2) α−1≠0α111α111α=0,α≠1R1→R1−R3,R2→R2→R3α−101−α0α−11−α11α=0α=1,1,−2 or α=1,−2

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$The value of | α | for which the system of equation$
$\begin{aligned} αx + y + z = α - 1 \\ x + αy + z = α - 1 \\ x + y + αz = α - 1 \end{aligned} has no solution, is_____________$

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answer is 2.

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Detailed Solution

System of equations
$\begin{aligned} αx + y + z = α - 1 \\ x + αy + z = α - 1 \\ x + y + αz = α - 1 \end{aligned}$
Since system has no solution.
$Therefore, (1) Δ = 0 and (2) α - 1 \neq 0$
$\begin{array}{l} |\begin{array}{ccc} α & 1 & 1 \\ 1 & α & 1 \\ 1 & 1 & α \end{array}| = 0, α \neq 1 \\ R_{1} \to R_{1} - R_{3}, R_{2} \to R_{2} \to R_{3} \\ |\begin{array}{ccc} α - 1 & 0 & 1 - α \\ 0 & α - 1 & 1 - α \\ 1 & 1 & α \end{array}| = 0 \\ α = 1, 1, - 2 or α = 1, - 2 \end{array}$