The set of all values of $\lambda$for which the system of linear equations  $2x_1-2x_2+x_3=\lambda x_1$, $2x_1-3x_2+2x_3=\lambda x_2$ ,  $-x_1+2x_3=\lambda x_3$ has a non-trivial solution

detailed solution

Correct option is C

The condition for a homogeneous system of equations to have a non-trivial solution is Δ=0⇒2−λ    −2          12         −3−λ      2−1             2            −λ=0      R1→R1+R3,R2→R1+2R3⇒1−λ         0            1−λ  0             1−λ       2−2λ−1                     2                     −λ=0 ⇒1−λ21         0             10       1             2−1      2      −λ=0                    C3→C3+C1⇒1−λ21         0         00          1          2−1       2    −λ+1=0⇒1−λ2−λ+1−4=0 ⇒λ=1,−3