# Question about determining the Moore-Penrose Inverse

can anybody help me? I'm wondering how you determine the Moore-Penrose inverse of the following matrix for example:

$A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix}$

The Moore-Penrose-Inverse $A^+$ must satisfy

$$A\cdot A^+ = I$$ i.e.,

$$\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix}\cdot \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$

Thus,

$$a+2e = 1$$ $$b+2f = 0$$ $$c+e = 0$$ $$d+f = 1$$ Following, $$a-2c = 1 \Leftrightarrow a = 1+2c$$ $$b-2d = -2 \Leftrightarrow b = -2+2d$$ $$e=-c$$ $$f=1-d$$

So the inverses are

$$A^+=\begin{bmatrix} 1 +2c& -2+2d \\ c & d \\ -c & 1-d \end{bmatrix}$$