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  4. The system of linear equations x+y-z=2, 2x+y-z=3, 3x + 2y + kz = 4 has a unique solution, if k is not equal to

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Q. The system of linear equations $x+y-z=2$, $2x+y-z=3$, $3x + 2y + kz = 4$ has a unique solution, if $k$ is not equal to

Determinants

Solution:

The system has a unique solution, if
given $\left|\begin{matrix}1&1&1\\ 2&1&-1\\ 3&2&k\end{matrix}\right|\ne0$
$\Rightarrow \quad1\left(k + 2 \right)-1\left(2 k + 3\right) + 1 \left( 4 - 3 \right) \ne 0$
$\Rightarrow \quad k + 2 - 2k - 3 + 1 \ne 0 \, \Rightarrow \quad k \ne0$