# Transpose

The transpose of a matrix A is another matrix where the rows of A are written as columns. Vectors can be transposed in the same way. We can write the transpose of A using different symbols such as AT [1][2], A[3], Atr and At.

## Examples

Here is the vector ${\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}}$ being transposed:

• ${\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1\\2\end{bmatrix}}.}$

Here are a few matrices being transposed:

• ${\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3\\2&4\end{bmatrix}}.}$
• ${\displaystyle {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}.\;}$
• ${\displaystyle {\begin{bmatrix}1&2&8\\3&4&3\\5&6&1\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3&5\\2&4&6\\8&3&1\end{bmatrix}}.\;}$

## Properties

Given two matrices A and B, the following properties related to the transpose are true:[3]

• ${\displaystyle (A^{T})^{-1}=(A^{-1})^{T}}$
• ${\displaystyle (AB)^{T}=B^{T}A^{T}}$

## References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-08.
2. Nykamp, Duane. "The transpose of a matrix". Math Insight. Retrieved September 8, 2020.
3. Weisstein, Eric W. "Transpose". mathworld.wolfram.com. Retrieved 2020-09-08.