If the system of linear equations 2x + 3y - z = - 2 x + y + z = 4 [x - y + | |z = 4 - 4] w…

Question

If the system of linear equations

$2x + 3y - z =  - 2$

$x + y + z = 4$

$$x - y + |\lambda |z = 4\lambda  - 4$$

where, $$\lambda$$ $$\in$$ R, has no solution, then

Accepted Answer

$$\Delta  = \left| {\begin{pmatrix} 2 & 3 & { - 1}  \
   1 & 1 & 1  \
   1 & { - 1} & {|\lambda |}  \

\end{pmatrix} } \right| = 0 \Rightarrow |\lambda, = 7$$

But at $$\lambda  = 7,\,{D_x} = {D_y} = {D_z} = 0$$

${P_1}:2x + 3y - z =  - 2$

${P_2}:x + y + z = 4$

$${P_3}:x - y + |\lambda |z = 4\lambda  - 4$$

So clearly $5{P_2} - 2{P_1} = {P_3}$, so at $$\lambda  = 7$$, system of equation is having infinite solutions.

So $$\lambda  =  - 7$$ is correct answer.

Solutions · 1