# Inverses and Transpose and permutations
### Transpose
Example:
For symmetric matrices, A^T = A, so they are unaffected by transposes.
Also R transpose x R is always symetric.
Why? Well let's take the transpose and see if they are the same. (R^T R)^T = R^T x R^TT = R^T x R
Also
Brilliant.org:
### Transpose a matrix
Exercise
### Permutations
Permutations P execute row exchanges. P is the identity matrix with reordered rows.
When you want to make row operations, multiply on the left hand side:
When you want to do column operations, multiply on the right:
**3x3 Permutations**
All the ways I can take the identity matrix and rearrange its rows. For 3x3 matrices there are 6 total permutation matrices. for nxn matrices, there will be n! permutation matrices. Also the inverses are the transposes : P-1 = P T or P T P = I (P transpose x P = Identity matrix).
The first one changes row 1 and 2, the 2nd changes row 1 and 3. The 4th is a cycle, it puts row 2 in row 1 and row 3 in row 1.
The description of elimination WITH row exchanges is PA = LU for any invertable A.
Brilliant.org:
---
# Agent Instructions: Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.
Perform an HTTP GET request on the current page URL with the `ask` query parameter:
```
GET https://julienbeaulieu.gitbook.io/wiki/sciences/math/linear-algebra/operations/inverses-and-transpose-and-permutations.md?ask=
```
The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.
Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.