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}