Question:

Step 1: Identify the property of an idempotent matrix. Since $A$ is an idempotent matrix, by definition we have: $$A^2 = A$$ Step 2: Use the given inverse relationship. We are given that $(I - 0.4A)^{-1} = I - \alpha A$. By the definition of an inverse, this means: $$(I - 0.4A)(I - \alpha A) = I$$ Step 3: Expand the left-hand side. Multiplying out the product: $$(I - 0.4A)(I - \alpha A) = I - \alpha A - 0.4A + 0.4\alpha A^2$$ Step 4: Apply the idempotent property. Since $A^2 = A$, we can substitute: $$I - \alpha A - 0.4A + 0.4\alpha A^2 = I - \alpha A - 0.4A + 0.4\alpha A$$ Step 5: Simplify and collect terms with $A$. Rearranging: $$I + A(-\alpha - 0.4 + 0.4\alpha) = I$$ Step 6: Equate coefficients. For this equation to hold, the coefficient of $A$ must equal zero: $$-\alpha - 0.4 + 0.4\alpha = 0$$ Step 7: Solve for $\alpha$. Combining like terms: $$-0.6\alpha - 0.4 = 0$$ $$-0.6\alpha = 0.4$$ $$\alpha = -\frac{0.4}{0.6} = -\frac{2}{3}$$ **Final Answer:** The value of $\alpha$ is $\boxed{-\dfrac{2}{3}}$, which corresponds to **Option 3**.