Matrices and Determinants 4 Question 15
16. The set of all values of $\lambda$ for which the system of linear equations $2 x _1-2 x _2+x _3=\lambda x _1, 2 x _1-3 x _2+2 x _3=\lambda x _2$ and $-x _1+2 x _2=\lambda x _3$ has a non-trivial solution
(2015 Main)
(a) is an empty set
(b) is a singleton set
(c) contains two elements
(d) contains more than two elements
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Answer:
Correct Answer: 16. (c)
Solution:
- Given system of linear equations
$$ \begin{aligned} 2 x _1-2 x _2+x _3 & =\lambda x _1 \\ \Rightarrow \quad(2-\lambda) x _1-2 x _2+x _3 & =0 \\ 2 x _1-3 x _2+2 x _3 & =\lambda x _2 \\ \Rightarrow \quad 2 x _1-(3+\lambda) x _2+2 x _3 & =0 \\ -x _1+2 x _2 & =\lambda x _3 \\ \Rightarrow \quad-x _1+2 x _2-\lambda x _3 & =0 \end{aligned} $$
Since, the system has non-trivial solution.
$$ \begin{array}{rlrlrl} & & 2-\lambda & -2 & 1 \\ & \therefore & 2 & -(3+\lambda) & 2 & =0 \\ & & \lambda & \lambda \end{array} $$
Hence, $\lambda$ contains two elements.
