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Question
The system of equations:
x + y + z = 5
x + 2y + 3z = 9
x + 3y + λz = µ
has a unique solution, if
(a) λ = 5, µ = 13
(b) λ ≠ 5
(c) λ = 5, µ ≠ 13
(d) µ ≠ 13
Options
λ = 5, µ = 13
λ ≠ 5
λ = 5, µ ≠ 13
µ ≠ 13
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Solution
\[(b) \lambda \neq 5\]
\[\text{ For a unique solution,}\left| A \right|\neq 0.\]
\[ \Rightarrow \begin{vmatrix}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda\end{vmatrix} \neq 0\]
\[ \Rightarrow 1\left( 2\lambda - 9 \right) - 1\left( \lambda - 3 \right) + 1\left( 3 - 2 \right) \neq 0\]
\[ \Rightarrow 2\lambda - 9 - \lambda + 3 + 1 \neq 0\]
\[ \Rightarrow \lambda - 5 \neq 0\]
\[ \Rightarrow \lambda \neq 5\]
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