1. **Problem Statement:** Verify the matrix identity $$(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$$ where $A$, $B$, and $C$ are invertible matrices. 2. **Recall the formula for the inverse of a product:** For any invertible matrices $X$ and $Y$, we have $$(XY)^{-1} = Y^{-1} X^{-1}.$$ This means the inverse of a product reverses the order of multiplication. 3. **Apply the formula step-by-step:** - Consider the product $ABC$. - Using the formula for two matrices, treat $AB$ as one matrix and multiply by $C$: $$ (ABC)^{-1} = ( (AB) C )^{-1} = C^{-1} (AB)^{-1} $$ - Now apply the formula again to $(AB)^{-1}$: $$ (AB)^{-1} = B^{-1} A^{-1} $$ 4. **Combine the results:** $$ (ABC)^{-1} = C^{-1} B^{-1} A^{-1} $$ 5. **Explanation:** The inverse of a product of matrices is the product of their inverses in reverse order. This is because matrix multiplication is associative but not commutative, so order matters. 6. **Conclusion:** The identity $$(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$$ is verified by applying the inverse of product rule twice. **Final answer:** $$(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$$