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 PT or P’, where P is any matrix.
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Transpose of a matrix example
Let R is a matrix such that
![\[R=\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-388763e038e451b281e3a78712ee09f2_l3.png "Rendered by QuickLaTeX.com")
Now, to find the transpose of this matrix R, we change rows into columns and columns into rows as follows.
![\[R^T=\begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-de8b127e4928f0d1d3a4a208a253e08e_l3.png "Rendered by QuickLaTeX.com")
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
![\[A=\begin{bmatrix} 4 \\ 9 \\ 5 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-1c5203080ba6b72e46e1802795a2214e_l3.png "Rendered by QuickLaTeX.com")
Now the transpose of matrix A is
![\[A^T=\begin{bmatrix} 4 & 9 & 5 \end{bmatrix}\]](https://electricalvoice.com/wp-content/ql-cache/quicklatex.com-3a301faaca8ce549eb2fc464d04b264b_l3.png "Rendered by QuickLaTeX.com")
Transpose of a matrix properties
The transpose of matrices P, Q and R are PT, QT and RT, respectively. Then
1. (PT)T = P, (QT)T = Q and (RT)T = R
2. (P + Q + R)T = PT + QT + RT
3. (PQR)T = RTQTPT
4. (PQ)T = QTPT
5. (kP)T = kPT, where k is a constant.
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