A matrix obtained as a resultant by changing rows into columns and columns into rows of any matrix is known as the **transpose of a matrix**. It is generally denoted by P^{T} or P’, where P is any matrix.

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## Transpose of a matrix example

Let R is a matrix such that

\[R=\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}\]

Now, to find the transpose of this matrix R, we change rows into columns and columns into rows as follows.

\[R^T=\begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}\]

This is the transpose of a 3 x 2 matrix R.

Let us take another example.

We have to find the transpose of matrix A such that

\[A=\begin{bmatrix} 4 \\ 9 \\ 5 \end{bmatrix}\]

Now the transpose of matrix A is

\[A^T=\begin{bmatrix} 4 & 9 & 5 \end{bmatrix}\]

## Transpose of a matrix properties

The transpose of matrices P, Q and R are P^{T}, Q^{T} and R^{T}, respectively. Then

1. (P^{T})^{T} = P, (Q^{T})^{T} = Q and (R^{T})^{T} = R

2. (P + Q + R)^{T} = P^{T} + Q^{T} + R^{T}

3. (PQR)^{T} = R^{T}Q^{T}P^{T}

4. (PQ)^{T} = Q^{T}P^{T}

5. (kP)^{T} = kP^{T}, where k is a constant.

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