The number of real values of λ , such that the system of linear...

For the system of linear equations

$2x - 3y + 5z = 9$

$x + 3y - z = - 18$

$3x - y + \left({\lambda}^{2} - \left|\lambda\right|\right)z = 16$ to have no solutions,

$∆$ should be equal to $0$.

i.e. \Delta = \left|\begin{cases}2 & - 3 & 5 \\ 1 & 3 & - 1 \\ 3 & - 1 & {\lambda}^{2} - \left|\lambda\right|\end{cases}\right|

$= 2\left(3{\lambda}^{2} - 3\left|\lambda\right| - 1\right) + 3\left({\lambda}^{2} - \left|\lambda\right| + 3\right) + 5\left(- 1 - 9\right)$

$= 9{\lambda}^{2} - 9\left|\lambda\right| - 43$

$= 9{\left|\lambda\right|}^{2} - 9\left|\lambda\right| - 43$

Now $∆ = 0$ for $2$ values of \left|\lambda\right|$as for the above quadratic the value of determinant is greater than zero.</p><p>So,$2$values of$\lambda satisfy the system of equations to obtain no solution.