Question

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

Options

• k ≠ 0

• −1 < k < 1

• −2 < k < 2

•  k = 0

Answer 1

(a) k ≠ 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.