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Question
The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
Options
µ only
λ only
λ and µ both
neither λ nor µ
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Solution
(a) µ only
\[\text{ For a unique solution, }\left| A \right|\neq 0\]
\[ \Rightarrow \begin{vmatrix}1 & 1 & 1 \\ 5 & - 1 & \mu \\ 2 & 3 & - 1\end{vmatrix} \neq 0\]
\[ \Rightarrow 1\left( 1 - 3\mu \right) - 1\left( - 5 - 2\mu \right) + 1\left( 15 + 2 \right) \neq 0\]
\[ \Rightarrow 1 - 3\mu + 5 + 2\mu + 17 \neq 0\]
\[ \Rightarrow - \mu + 23 \neq 0\]
\[ \Rightarrow \mu \neq 23\]
\[\text{ So, existence of a unique solution depends only on }\mu.\]
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