Mathematics | TS EAMCET 2019 | Discussion Page

Consider the following system of equations in matrix form

$\begin{bmatrix}1 \\2 \\\lambda \end{bmatrix}$ (1 2 $\lambda$ ) $\begin{bmatrix}x \\y \\z\end{bmatrix} =0$

Then which one of the following statements is ture?

$\forall \lambda\epsilon(-\infty,\infty)$ , the given system has non trivial solution

$\forall \lambda\epsilon(-\infty,\infty)$ , the given system has only trivial solution

C) For

$\lambda\neq0$ , the given system does not have any solution

D) For

$\lambda =0$ , the given system is inconsistent

Option A

We have ,

$\begin{bmatrix}1 \\2 \\\lambda \end{bmatrix}$ (1 2 $\lambda$ ) $\begin{bmatrix}x \\y \\z\end{bmatrix} =0$

$\begin{bmatrix}1 & 2 &\lambda \\2 & 4 & 2\lambda \\\lambda &2\lambda &\lambda^{2} \end{bmatrix}\begin{bmatrix}x \\y \\z \end{bmatrix}=0$

Let A= $\begin{bmatrix}1 & 2 &\lambda \\2 & 4 & 2\lambda \\\lambda &2\lambda &\lambda^{2} \end{bmatrix}$

|A|= 0, $\forall \lambda \epsilon R$

$\therefore$ the given syste m has non trivial solution.

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