If $A$ and $B$ are invertible $n imes n$ matrices, then $(AB)^{-1}$ …

Question

If $A$ and $B$ are invertible $n 	imes n$ matrices, then $(AB)^{-1}$ equals:

Accepted Answer

B. $B^{-1} A^{-1}$

$(AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1} = I$, and similarly on the left. Therefore $(AB)^{-1} = B^{-1}A^{-1}$, with the order reversed (the 'shoes-and-socks' rule). Option C is just the formula for the inverse and does not contradict B.

Answer

$A^{-1} B^{-1}$

Answer

$(A B)^{-1} = \dfrac{1}{|AB|}\operatorname{adj}(AB)$ only

Answer

Does not exist in general

If $A$ and $B$ are invertible $n \times n$ matrices, then $(AB)^{-1}$ equals: