Inverse of 2x1 matrix.

**eulerian** · February 7th 2011, 01:07 AM

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

What is the inverse of A?

Thanks

**Prove It** · February 7th 2011, 01:15 AM

Only square matrices have inverses...

**eulerian** · February 7th 2011, 01:28 AM

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

**FernandoRevilla** · February 7th 2011, 01:46 AM

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 $R=(\alpha,\beta)$ satisfying

$\begin{pmatrix}{a}\\{b}\end{pmatrix} (\alpha,\beta)=\begin{pmatrix}{1}&{0}\\{0}&{1}\end {pmatrix}$

and about the existence of left inverse $L=(\lambda,\mu)^t$ satisfying

$(\lambda,\mu)\begin{pmatrix}{a}\\{b}\end{pmatrix}= (1)$

Fernando Revilla

**Ackbeet** · February 7th 2011, 02:36 AM

Originally Posted by FernandoRevilla

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

I don't think that a right inverse is possible. Consider the equations involved:

$a\alpha=1\not=0,$ which implies $a\not=0$ and $\alpha\not=0.$ Also,

$b\beta=1\not=0,$ implying $b\not=0$ and $\beta\not=0.$

However, $a\beta=0$ , which implies that either $a=0$ or $\beta=0,$ a contradiction.

Therefore, a right inverse doesn't exist in this case.

**FernandoRevilla** · February 7th 2011, 02:46 AM

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 $a=b=0$ does not exist $R$ and does not exist $L$ .

Fernando Revilla

**Ackbeet** · February 7th 2011, 02:53 AM

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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