# Talk:Inverse element

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Field: Algebra

## left/right matrix inverse

I think the phrase

If the determinant of M is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one.

is redundant, and not very clear, (IMHO):

1. why det=0 => no one-sided inverse ?
2. why "no one-sided inverse" => "l/r inverse implies existence of the other" ?
3. more specifically, why is a left inverse matrix also is a right inverse ?

But as I already did some "cutting", I'll leave it for the moment... MFH 15:17, 5 Apr 2005 (UTC)

I disagree...

${\displaystyle {\begin{bmatrix}17&22&27\\22&29&36\\27&36&45\end{bmatrix}}}$

The statement above that this "Is a singular matrix, and can't be inverted." is partly incorrect. An inverse is shown here:

${\displaystyle {\begin{bmatrix}1.25&0.0&-0.75\\0.0&0.0&0.0\\-0.75&0.0&0.47222\end{bmatrix}}}$

This matrix is a left and right hand inverse (Moore-Penrose properties 1 and 2).

The null-space for this matrix is:

${\displaystyle {\begin{bmatrix}0.0&-0.5&0.0\\0.0&1.0&0.0\\0.0&-0.5&0.0\end{bmatrix}}}$

There are an infinite number of solutions to the underspecified or singular matrix. The calculations are not difficult and are described on the http://mjollnir.com/matrix/demo.html page. Should I create a wikipedia page to describe the algorithm?

The original matrix was:

${\displaystyle {\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}}$

Using the http://mjollnir.com/matrix/demo.html site online calculator, a pseudo-inverse is directly calculated:

${\displaystyle {\begin{bmatrix}-1.0&0.5\\0.0&0.0\\0.666667&-0.166667\end{bmatrix}}}$

The null space is the same as the above:

${\displaystyle {\begin{bmatrix}0.0&-0.5&0.0\\0.0&1.0&0.0\\0.0&-0.5&0.0\end{bmatrix}}}$

rand huso (talk) 18:08, 23 February 2008 (UTC)