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The set of all values of $\lambda$ for which the system of linear equations

$x-2y-2z=\lambda x$

$x+2y+z=\lambda y$

$-x-y=\lambda z$

has a non-trivial solution :
Option 1)

contains more than two elements
Option 2)

is a singleton
Option 3)

is an empty set
Option 4)

contains exactly two elements

Answers (1)

best_answer

Solution of a homogeneous system of linear equations -

Let $Ax=0$

If $A$ is singular then the system of equations will have infinitely many solutions

-

Cramer's rule for solving system of linear equations -

When $\Delta =0$ and $\Delta _{1}=\Delta _{2}=\Delta _{3}=0$ ,

then the system of equations has infinite solutions.

- wherein

$a_{1}x+b_{1}y+c_{1}z=d_{1}$

$a_{2}x+b_{2}y+c_{2}z=d_{2}$

$a_{3}x+b_{3}y+c_{3}z=d_{3}$

and

$\Delta =\begin{vmatrix} a_{1} &b_{1} &c_{1} \\ a_{2} & b_{2} &c_{2} \\ a_{3}&b _{3} & c_{3} \end{vmatrix}$

$\Delta _{1},\Delta _{2},\Delta _{3}$ are obtained by replacing column 1,2,3 of $\Delta$ by $\left ( d_{1},d_{2},d_{3} \right )$ column

For non-trivial solution , $\Delta =0$

$\Rightarrow \begin{vmatrix} 1-\lambda &-2 &-2 \\ 1& 2-\lambda &1 \\ -1&-1 & -\lambda \end{vmatrix}=0$

$\Rightarrow \lambda=1\left\: ( singleton \right )$

Option 1)

contains more than two elements

Option 2)

is a singleton
Option 3)

is an empty set

Option 4)

contains exactly two elements

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