Question

Show that if A and B are invertible matrices and AB exists, then$(AB{)}^{-1}=B^{-1}{A}^{-1}$

Short answer

If A and B are invertible matrices and AB exists, then$(AB{)}^{-1}=B^{-1}{A}^{-1}$

Step-by-step solution

Step: 1

Given A and B are invertible matrices.

From the definition of inverse of matrix (AB)

$$\begin{gathered} (AB{)}^{-1}=1 \\ {A}^{-1}(AB\left)\right(AB{)}^{-1}=A^{-1} \\ \left(A^{-1}A\right)B(AB{)}^{-1}=A^{-1} \\ IB(AB{)}^{-1}=A^{-1} \\ B(AB{)}^{-1}=A^{-1} \\ {B}^{-1}B(AB{)}^{-1}=B^{-1}{A}^{-1} \\ I(AB{)}^{-1}=B^{-1}{A}^{-1} \\ (AB{)}^{-1}=B^{-1}{A}^{-1} \end{gathered}$$