$\displaystyle A=\left(\begin{array}{cc}a\\b\end{array}\right)$
What is the inverse of A?
Thanks
You can speak about the existence of right inverse $\displaystyle R=(\alpha,\beta)$ satisfying
$\displaystyle \begin{pmatrix}{a}\\{b}\end{pmatrix} (\alpha,\beta)=\begin{pmatrix}{1}&{0}\\{0}&{1}\end {pmatrix}$
and about the existence of left inverse $\displaystyle L=(\lambda,\mu)^t$ satisfying
$\displaystyle (\lambda,\mu)\begin{pmatrix}{a}\\{b}\end{pmatrix}= (1)$
Fernando Revilla
I don't think that a right inverse is possible. Consider the equations involved:
$\displaystyle a\alpha=1\not=0,$ which implies $\displaystyle a\not=0$ and $\displaystyle \alpha\not=0.$ Also,
$\displaystyle b\beta=1\not=0,$ implying $\displaystyle b\not=0$ and $\displaystyle \beta\not=0.$
However, $\displaystyle a\beta=0$, which implies that either $\displaystyle a=0$ or $\displaystyle \beta=0,$ a contradiction.
Therefore, a right inverse doesn't exist in this case.
I said, we can speak about the existence of right and left inverse (i.e. it has sense to define them). Of course left and/or right inverse could not exist.
Choosing for example $\displaystyle a=b=0$ does not exist $\displaystyle R$ and does not exist $\displaystyle L$.
Fernando Revilla