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