1. **State the problem:** We need to find the value of $A$ that satisfies the equation $$2 \frac{1}{A} = \frac{19}{A}.$$ 2. **Rewrite the mixed fraction:** The mixed fraction $2 \frac{1}{A}$ can be written as an improper fraction: $$2 \frac{1}{A} = \frac{2A + 1}{A}.$$ 3. **Set up the equation:** Now the equation becomes $$\frac{2A + 1}{A} = \frac{19}{A}.$$ 4. **Multiply both sides by $A$ to eliminate denominators:** $$\cancel{A} \times \frac{2A + 1}{\cancel{A}} = \cancel{A} \times \frac{19}{\cancel{A}} \implies 2A + 1 = 19.$$ 5. **Solve for $A$:** $$2A + 1 = 19$$ $$2A = 19 - 1$$ $$2A = 18$$ $$A = \frac{18}{2}$$ $$A = 9.$$ **Final answer:** $A = 9$