The system of linear equations: x + y + z = 2 2x + y − z = 3 3x + 2y + kz = 4 has a unique solution if

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(a) k $\neq$ 0 \text{ For a unique solution, }\left| A \right|\neq 0. The given system of equations can be written in matrix form as follows: \begin{bmatrix}1 & 1 & 1 \ 2 & 1 & - 1 \ 3 & 2 & k\end{bmatrix}\neq0 \Rightarrow 1\left( k + 2 \right) - 1\left( 2k + 3 \right) + 1\left( 4 - 3 \right)\neq0 \Rightarrow k + 2 - 2k - 3 + 1\neq 0 \Rightarrow k \neq 0 So, the given system of equations has a unique solution ifkis not equal to 0.

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