Finding Pseudoinverse (Moore-Penrose) through the One-Sided Inverses

[Eobseo]

Homework Statement

I want to calculate the (unique) Moore-Penrose Pseudoinverse by knowing the one-sided inverses of a matrix

Homework Equations

Consider a matrix such as $$B = \begin{bmatrix}
1 & 0 & 2 \\
0 & 1 & 1
\end{bmatrix}$$ I know how to compute the right inverses (or in the case of $m\geq n$ the left inverses) and have done so; I've obtained the result $$B^{-1}_{R} = \begin{bmatrix}
1-2c_{1} & -2c_{2} \\
-c_{1} & 1-c_{2} \\
c_{1} & c_{2}
\end{bmatrix}$$.

The Attempt at a Solution

However, I now want to calculate the (unique) Moore-Penrose Pseudoinverse, preferably using this right-sided inverse. Clearly, it would have to be one of the right-sided inverses. Using MATLAB I've found that the Moore-Penrose Pseudo inverse equals $B_{R}^{-1}$ for $c_{1} = \frac{1}{3}, c_{2} = \frac{1}{6}$. Is there a way I could easily get those correct values of $c_{1}$ and $$c_{2}## (by that I mean without using a tool such as Matlab), as in, find those corresponding to the Moore-Penrose Pseudoinverse?

 

[haruspex]

There's a bunch of methods at http://en.wikipedia.org/wiki/Moore–Penrose_pseudoinverse#Construction