1. **Problem statement:** Simplify the expression $$AB (A^{-1}B)^{-1}$$ where $A$ and $B$ are non-singular square matrices. 2. **Recall the properties:** - The inverse of a product of matrices satisfies $$(XY)^{-1} = Y^{-1}X^{-1}$$. - Since $A$ and $B$ are non-singular, their inverses $A^{-1}$ and $B^{-1}$ exist. 3. **Apply the inverse property:** $$(A^{-1}B)^{-1} = B^{-1}(A^{-1})^{-1} = B^{-1}A$$ 4. **Substitute back into the expression:** $$AB (A^{-1}B)^{-1} = AB (B^{-1}A)$$ 5. **Simplify the product:** $$AB B^{-1} A = A (B B^{-1}) A = A I A = A^2$$ 6. **Final answer:** $$AB (A^{-1}B)^{-1} = A^2$$