Transpose of a Matrix – Properties

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.

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 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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