Question

In the derivation of \[(AB)^{-1}=B^{-1}A^{-1}\], which equality changes \[A^{-1}(AB)(AB)^{-1}=A^{-1}\] into \[B(AB)^{-1}=A^{-1}\]?

Options

• \[BB^{-1}=I\]

• \[AA^{-1}=I\]

• \[A^{-1}A=I\]

• \[B^{-1}B=I\]

Answer 1

Associating the left side gives \((A^{-1}A)B(AB)^{-1}\). Since \(A^{-1}A=I\), this becomes \(IB(AB)^{-1}=B(AB)^{-1}\).