- #1

ELB27

- 117

- 15

## Homework Statement

Let ##P## be a permutation matrix. Show that for some ##N>0## [tex]P^N := \underbrace{PP...P}_{N \ \text{times}} = I[/tex]

**2. Relevant definitions**

A permutation matrix is a ##n\times n## matrix containing only zeros and ones such that there is exactly one ##1## per row and per column.

## The Attempt at a Solution

Well, given the above definition, this permutation matrix is simply the result of column switching of the identity matrix. So, ##P## can be expressed as ##P=IE=E## where ##E## is an elementary matrix performing column switching. So, the problem reduces to proving that after some finite number of column switching of the identity matrix we get it back. However, I have no idea how to formally prove it - it seems plausible (there is only a finite number of permutations possible for those columns) but that is not formal.

So, how to formulate such a proof in a formal manner?

Any help is greatly appreciated!