AS October 2021 Paper 1 Q4

4 Anika thinks that, for two square matrices \(\mathbf{A}\) and \(\mathbf{B}\), the inverse of \(\mathbf{AB}\) is \(\mathbf{A}^{-1}\mathbf{B}^{-1}\). Her attempted proof of this is as follows.

\[\begin{aligned} (\mathbf{AB})(\mathbf{A}^{-1}\mathbf{B}^{-1}) &= \mathbf{A}(\mathbf{BA}^{-1})\mathbf{B}^{-1} \\ &= \mathbf{A}(\mathbf{A}^{-1}\mathbf{B})\mathbf{B}^{-1} \\ &= (\mathbf{AA}^{-1})(\mathbf{BB}^{-1}) \\ &= \mathbf{I} \times \mathbf{I} \\ &= \mathbf{I} \end{aligned}\]

Hence \((\mathbf{AB})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\)