Question

What does transpose mean?

Answer 1

Hint: As we know that above question is related to matrix. We know that a matrix is a collection of numbers, which are organised in rows and columns. Or we can say that it is an array of numbers arranged in a rectangular way and they are divided between rows and columns. We can say that the transpose of matrix can be written as the matrix appeared by rearranging the rows and columns i.e. rows as columns and columns as rows.

Complete answer:
We can say that the transpose of the matrix is received by rearranging the rows and columns in the matrix i.e. the elements on the transpose change their position but the values remain the same. Each and every matrix can have the transpose. We can denote the transpose of the matrix $A$ by ${A^T}$. As we can see it by the example, Let matrix $A = \left( {\begin{array}{*{20}{c}}
  1&2&3 \\
  4&5&6
\end{array}} \right)$. So we can write the transpose of matrix $A$ as ${A^T} = \left( {\begin{array}{*{20}{c}}
  1&4 \\
  2&5 \\
  3&6
\end{array}} \right)$. Hence the transpose of a rectangular matrix is also a rectangular matrix.

Note: We should note all the important properties of matrices and the transpose as they are very useful while solving questions. We should note the property that if a matrix has $m \times n$ dimensions then the transpose of the matrix will have the dimensions as $n \times m$. The superscript written in the ${A^T}$, $T$means the transpose of the matrix.