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 P^T 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 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^TQ^TP^T

4. (PQ)^T = Q^TP^T

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

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