# 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...

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

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

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

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

The null-space for this matrix is:

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

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:

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

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

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

The null space is the same as the above:

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

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