The greatest value of c\; \epsilon \; \mathbb{R} for which the system of linear equations

x-cy-cz=0

cx-y+cz=0

cx+cy-z=0

has a non-trivial solution, is :
 

  • Option 1)

    \frac{1}{2}

  • Option 2)

    0

  • Option 3)

    2

     

  • Option 4)

    -1

Answers (1)

For non trivial solution \Delta =0

\begin{vmatrix} 1 & -c &-c \\ c&-1 &c \\ c& c &-1 \end{vmatrix}=1(1-c^{2})+c(-\epsilon -c^{2})-c(c^{2}+c)=0 

                             

=1-c^{2}-c^{2}-c^{3}-c^{3}-c^{2}=0

=-3c^{2}-2c^{3}+1=0

=(c+1)^2(c-\frac{1}{2})=0

c=-1,\frac{1}{2}\; \; \; \frac{1}{2}>-1

\frac{1}{2} \; is \; \; greatest

So \: C=\frac{1}{2}


Option 1)

\frac{1}{2}

Option 2)

0

Option 3)

2

 

Option 4)

-1

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