The value of $\lambda $ for which the given system of equations will have infinitely many solutions $x+y -6z =0; \lambda x -y +2z=0;-3x+\lambda y +2z=0,$ is /are :

$\lambda = -\frac{5}{3}$ or 1

The correct answer is Option (3) → $\lambda = -\frac{5}{3}$ or 1

for infinite solutions

$\begin{vmatrix}1&1&-6\\λ&-1&2\\-3&λ&2\end{vmatrix}=0$

$C_2→C_2-C_1$

$C_3→C_3+6C_1$

$\begin{vmatrix}1&0&0\\λ&-1-λ&2+6λ\\-3&λ+3&-16\end{vmatrix}=0$

$16(1+λ)-2(λ+3)^2=0$

or $(λ+3)^2-8(λ+1)=0⇒λ^2+6λ+9-8λ-8=0$

$⇒λ^2-2λ+1=0$

$16(1+λ)-2(λ+3)(1+3λ)=0$

$8+8λ=3λ^2+3+10λ$

$3λ^2-5+2λ=0$

$⇒3λ^2+5λ-3λ-5=0$

$(3λ+5)(λ-1)=0$

$λ=1,-\frac{5}{3}$