# Thread: Inverse of 2x1 matrix.

1. ## Inverse of 2x1 matrix.

$\displaystyle A=\left(\begin{array}{cc}a\\b\end{array}\right)$

What is the inverse of A?

Thanks

2. Only square matrices have inverses...

3. Can I partition the matrix and then directly reverse a and b separately, although this seems odd?

4. Originally Posted by eulerian
Can I partition the matrix and then directly reverse a and b separately, although this seems odd?

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

5. Originally Posted by FernandoRevilla
You can speak about the existence of right inverse $\displaystyle R=(\alpha,\beta)$ satisfying
$\displaystyle \begin{pmatrix}{a}\\{b}\end{pmatrix} (\alpha,\beta)=\begin{pmatrix}{1}&{0}\\{0}&{1}\end {pmatrix}$
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.

6. Originally Posted by Ackbeet
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

7. Originally Posted by FernandoRevilla
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.

Fernando Revilla
Got it.

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