College Algebra (6th Edition)

The statement is false. To make it true, replace "$=A^{-1}B^{-1}$" with " $=B^{-1}A^{-1}$ ".
Since matrix multiplication is associative, $(AB)(B^{-1}A^{-1})=A(BB^{-1})A^{-1}$ $=AIA^{-1}=AA^{-1}=I$ and $(B^{-1}A^{-1})(AB)=B^{-1}(A^{-1}A)B$ $=B^{-1}IB=B^{-1}B=I$, the inverse of $(AB)$ is $(B^{-1}A^{-1})$, not $(A^{-1}B^{-1}).$ The statement is false. To make it true, replace "$=A^{-1}B^{-1}$" with " $=B^{-1}A^{-1}$ ".