Left/right inverse matrix question For what values of $a,b,c$ does a left and/or right inverse exist for $A=\begin{bmatrix}
1 & a \\
2 & b \\
3 & c
\end{bmatrix}$ exist?
We know that a left inverse matrix $X$ exists such that $XA=I_2$ where $I_2$ is the $2\times 2$ identity matrix so $X$ is a $2\times 3$ matrix. What do we do next? Thanks.

Here we can view $A$ as a map from $\mathcal{R}^2$ to $\mathcal{R}^3$.
Having a left inverse is impossible if $A$ is not an one to one mapping.
Image of the mapping $A$ is the column space of $A$. 
If the columns space is 1-dimentional then $A$ is not one to one (because by rank-nullity theorem, kernel of $A$ is no trivial). 
So to have a left inverse we need $[a,b,c]$ not a multiple of $[1,2,3]$ to guaranty that image of $A$ is not 1-dimentional.
Having a right inverse is impossible as there are no mapping $X$ that is one to one from $\mathcal{R}^3$ to $\mathcal{R}^2$.
