1. **State the problem:** Simplify the expression $$\left(\frac{a^{-1}}{-ab \cdot 2a^3}\right)^2$$. 2. **Rewrite the denominator:** The denominator is $$-ab \cdot 2a^3 = -2ab a^3$$. 3. **Combine like terms in the denominator:** Since $$a^1 \cdot a^3 = a^{1+3} = a^4$$, the denominator becomes $$-2b a^4$$. 4. **Rewrite the fraction:** $$\frac{a^{-1}}{-2b a^4} = \frac{a^{-1}}{-2b a^4}$$ 5. **Combine powers of $$a$$ in numerator and denominator:** $$= \frac{a^{-1}}{a^4} \cdot \frac{1}{-2b} = \frac{a^{-1-4}}{-2b} = \frac{a^{-5}}{-2b}$$ 6. **Rewrite the fraction:** $$= \frac{1}{-2b a^5}$$ 7. **Square the entire fraction:** $$\left(\frac{1}{-2b a^5}\right)^2 = \frac{1^2}{(-2b)^2 (a^5)^2} = \frac{1}{4 b^2 a^{10}}$$ **Final answer:** $$\boxed{\frac{1}{4 b^2 a^{10}}}$$