Transpose of a Matrix

In general understanding, ‘trans’ means to change and ‘pose’ means position. Transpose of a matrix is obtained by inter-changing the position of its rows and columns.

Formula for Transpose of Matrix

Let $\text{A = [a}_{ij}\text{]}$ be a matrix of order mxn then the transpose of the matrix is a new matrix $\text{A}^{T} = [\text{a}_{ji}]$ of order nxm.

During the transpose of a matrix, the elements on the position of the elements on the principal diagonal do not change.

Transpose of a 1×1 Matrix

Let

$\text{A} = \left ( a_{11} \right)_{1 \text{x} 1}$

Then the tranpose of matrix A is written as

$\text{A} = \left ( a_{11} \right)_{1 \text{x} 1}$

NOTE: The tranpose of a 1×1 square matrix is the matrix itself.

Transpose of a 2×2 Matrix

Let

$\text{A} = \left ( \displaylines{a_{11} & a_{12} \\ a_{21} & a_{22} } \right )_{2\text{x}2}$

Then the transpose of matrix A is written as

$\text{A} = \left ( \displaylines{a_{11} & a_{21} \\ a_{12} & a_{22} } \right )_{2\text{x}2}$

Transpose of a 3×3 Matrix

Let

$\text{A} = \left ( \displaylines{a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}} \right )_{3\text{x}3}$

Then the transpose of matrix A is written as

$\text{A}^\text{T} = \left ( \displaylines{ a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} } \right)_{3\text{x}3} $

Applications of Tranpose of Matrices

Tranpose of matrices is used in finding the adjoint of the given matrix which allows us to find the inverse of the given matrix. Hence, tranpose of matrix is used in finding the inverse of a given matrix.