First slide

System of linear equations

Question

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

Moderate

A

-2
B

either -2 or 1
C

not -2
D

1

Solution

$T h e c o n d i t i o n f o r n o n - e x i s \tan c e o f s o l u t i o n i s Δ = 0 \Rightarrow |\begin{matrix} α & 1 & 1 \\ 1 & α & 1 \\ 1 & 1 & α \end{matrix}| = 0$
$\Rightarrow α (α^{2} - 1) - 1 (α - 1) + 1 (1 - α) = 0$
$\Rightarrow α^{3} - 3 α + 2 = 0 \Rightarrow α = - 2, 1$
If $α = 1,$ all are coincident
It has infinitely many solutions
$∴ α = - 2$

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