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Transpose of a Matrix

Tool to compute the transpose of a matrix. The transpose of a matrix M of size mxn is a matrix denoted tM of size nxm created by swapping lines and columns.

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Transpose of a Matrix -

Tag(s) : Matrix

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# Transpose of a Matrix

## Matrix Transpose Calculator NxN

### How to calculate the transpose of a matrix?

The transposition of a matrix (or transpose of matrix) is one of the most basic matrix operations to perform. The transpose of a matrix consists of inverting the rows with the columns:

$$\text{ If } M = \begin{bmatrix} a & c & e \\ b & d & f \end{bmatrix} \text{ Then } M^T = \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}$$

The lines are read from left to right and are transposed from top to bottom.

Example: $$M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \Rightarrow M^t = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$$

The transposition of a matrix $M$ is noted $M^t$ or $^tM$. The transposition operation is then noted with an exponent T or t (uppercase or lowercase) prefixed or postfixed.

The transposition is valid on both square matrices and rectangular matrices. A transposed row vector is a column vector and vice versa.

### What is a double transposition?

Transposing twice a matrix returns it unchanged.

The double transposition is the name given to a cryptographic cipher.

### What is the transpose of a row matrix or a column matrix?

The transpose of a column matrix is a line matrix of the same size and vice versa.

Example: The transpose from $\begin{bmatrix} a \\ b \end{bmatrix}$ is $\begin{bmatrix} a & b \end{bmatrix}$

Example: The transpose from $\begin{bmatrix} a & b \end{bmatrix}$ is $\begin{bmatrix} a \\ b \end{bmatrix}$

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