Prove the following generalization of Exercise 6. If A is an \(m \times n\) matrix, B is an \(n \times m\) matrix and n < m, then AB is not invertible.
Solution
The columns of \(AB\) lie in the column space of \(A\). Hence \[ \operatorname{rank}(AB)\le \operatorname{rank}(A)\le n<m. \] But an invertible \(m\times m\) matrix has rank \(m\). Therefore \(AB\) cannot be invertible.