$$\Delta $$ = $$\left| {\matrix{ \lambda & 2 & 2 \cr {2\lambda } & 3 & 5 \cr 4 & \lambda & 6 \cr } } \right|$$

= $$\lambda $$ ( 18 – 5$$\lambda $$) – 2(12$$\lambda $$ – 20) + 2(2$$\lambda $$² – 12)

= 18$$\lambda $$ – 5$$\lambda $$² – 24$$\lambda $$ + 40 + 4$$\lambda $$² – 24

= – $$\lambda $$² – 6$$\lambda $$ + 16

= – ($$\lambda $$ + 8)($$\lambda $$ – 2)

For no solutions $$\lambda $$ = 0 $$ \Rightarrow $$ $$\lambda $$ = – 8, $$\lambda $$ = 2

when $$\lambda $$ = 2

$$\Delta $$_x = $$\left| {\matrix{ 5 & 2 & 2 \cr 8 & 3 & 5 \cr {10} & 2 & 6 \cr } } \right|$$

= 5 (18 – 10) – 2 (48 – 50) + 2 (16 – 30)

= 40 + 4 – 28 $$ \ne $$ 0

So no solution for $$\lambda $$ = 2