Let $I_M \in \mathbb{R}^{M \times M}$ be a D5621: Real identity matrix.
A D4571: Real matrix $A \in \mathbb{R}^{N \times M}$ is left-invertible if and only if \begin{equation} \exists \, B \in \mathbb{R}^{M \times N} : B A = I_M \end{equation}