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# 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 $\begin{bmatrix} 1 & 2 \end{bmatrix}$ being transposed:

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

Here are a few matrices being transposed:

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

## Properties

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

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

## References

1. Nykamp, Duane. "The transpose of a matrix".
2. Weisstein, Eric W.. "Transpose" (in en).
de:Matrix (Mathematik)#Die transponierte Matrix