Mathematics - Determinant Question with Solution | TestHub

Given, the system of linear equations, 

$x+y+z=4 ........................\left(i\right)$

$3x+2y+5z=3................\left(ii\right)$

$9x+4y+(28+[\lambda \left]\right]z=\left[\lambda \right].....................\left(iii\right)$

$D=\left|\begin{array}{ccc}1& 1& 1\\ 3& 2& 5\\ 9& 4& 28+\left[\lambda \right]\end{array}\right|$

$=1\left\{56+2\left[\lambda \right]-20\right\}-1\left\{84+3\left[\lambda \right]-45\right\}+1\left(12-18\right)$

$D=-\left(\right[\lambda ]+9)$

If $D=-\left(\left[\lambda \right]+9\right)\ne 0$ then unique solution
If $D=-\left(\left[\lambda \right]+9\right)=0$ then $D_1=D_2=D_3=0$ infinite solutions

Hence,  $\lambda \in R$