First slide

System of linear equations

Question

The system of equations
$k x + y + z = k, x + k y + z = k, x + y + k z = k$ has no solution , If K=

Moderate

A

1
B

2
C

$- 2$
D

$- 1$

Solution

Since $|\begin{matrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{matrix}| = 0 \Rightarrow k (k^{2} - 1) - 1 (k - 1) + 1 (1 - k) = 0$
$\Rightarrow k^{3} - k - k + 1 + 1 - k = 0$
$\Rightarrow k^{3} - 3 k + 2 = 0 \Rightarrow k = 1 (o r) - 2$
If K=1 Then the equations are coincident
$∴ k = - 2$

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