Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 326 / 1 / b / iv / Solution

The Reverse-order law for the Moore--Penrose inverse is false in general. TakeThenA sufficient condition isso that $A$ has full column rank and $B$ has full row rank. Indeed $A^{†}A=I_n$ and $B{B}^{†}=I_n$; these identities make $B^{†}{A}^{†}$ satisfy all four Penrose equations for $AB$.