The value of k for which, the system of equations
kx + 3y - z = 1; x + 2y + z = 2 and -kx + y + 2z = -1 have no solution, is .

k = -7

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k = 7

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k = - \frac{7}{2}

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k = \frac{7}{2}

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Solution

The correct option is C $k = - \frac{7}{2}$
The given system of equations will have no solution if:
$D = ⎛ ⎜ ⎝ \begin{matrix} k & 3 & - 1 1 & 2 & 1 - k & 1 & 2 \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow k (4 - 1) - 3 (2 + k) - 1 (1 + 2 k) = 0 \Rightarrow - 2 k - 7 = 0 \Rightarrow k = - \frac{7}{2} For, k = - \frac{7}{2}, D_{1} = ⎛ ⎜ ⎝ \begin{matrix} 1 & 3 & - 1 2 & 2 & 1 - 1 & 1 & 2 \end{matrix} ∣ ∣ ∣ ∣ = - 16 \neq 0 Hence, D = 0 and D_{1} \neq 0 . So, the given system of equations will have no solution for k = - \frac{7}{2}$

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