Matrices and Determinants 4 Question 6
6. The greatest value of $c \in R$ for which the system of linear equations $x-c y-c z=0, c x-y+c z=0$,
$c x+c y-z=0$ has a non-trivial solution, is
(a) -1
(b) $\frac{1}{2}$
(c) 2
(d) 0
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Answer:
Correct Answer: 6. (b)
Solution:
Key Idea: A homogeneous system of linear equations have non-trivial solutions if $\Delta=0$
Given system of linear equations is
$ \begin{aligned} & x-c y-c z=0 \\ & c x-y+c z=0 \end{aligned} $
and $ c x+c y-z=0 $
We know that a homogeneous system of linear equations have non-trivial solutions if
$ \begin{aligned} & \Delta=0 \\ & \Rightarrow \quad\left|\begin{array}{rrr} 1 & -c & -c \\ c & -1 & c \\ c & c & -1 \end{array}\right|=0 \\ & \Rightarrow 1\left(1-c^2\right)+c\left(-c-c^2\right)-c\left(c^2+c\right)=0 \\ & \Rightarrow \quad 1-c^2-c^2-c^3-c^3-c^2=0 \\ & \Rightarrow \quad-2 c^3-3 c^2+1=0 \\ & \Rightarrow \quad 2 c^3+3 c^2-1=0 \\ & \Rightarrow \quad(c+1)\left[2 c^2+c-1\right]=0 \\ & \Rightarrow \quad(c+1)\left[2 c^2+2 c-c-1\right]=0 \\ & \Rightarrow \quad(c+1)(2 c-1)(c+1)=0 \\ & \Rightarrow \quad c=-1 \text { or } \frac{1}{2} \\ & \end{aligned} $
Clearly, the greatest value of $c$ is $\frac{1}{2}$.
