Subsection 2.3.2 The Transpose of a Matrix, \(A^{t}\)
If the rows and columns of any matrix A is interchanged then the new matrix is known as transpose of the matrix A and it is denoted by \(A^{t}\) or \(A'\text{,}\) i.e. \(A=a_{ij}\) and \(A^{t}=a_{ji}\text{.}\) For example: if
\begin{equation*}
A = {\begin{bmatrix}
1 & -1 & 2 \\ 3 & 0 & 1 \\
\end{bmatrix}}
\end{equation*}
then,
\begin{equation*}
A^{t} = {\begin{bmatrix}
1 & 3 \\ -1 & 0 \\ 2 & 1
\end{bmatrix}}
\end{equation*}
Properties of Transpose Matrix.
- The transpose of the transpose of a matrix \(A\) is the matrix itself \(A\text{,}\) i.e., \((A^{t})^{t} = A\text{.}\)
- The transpose of the sum of two matrices is the sum of their transposes, i.e., \((A+B)^{t}=A^{t}+B^{t}\text{.}\)
- The transpose of a scalar times the matrix is the scalar times the transpose of the matrix, i.e., \((kA)^{t} = kA^{t}\) where \(k\) is a scalar.
- The transpose of the product of two matrices is the product in reverse order of their transpose, i.e., \((AB)^{t} = B^{t}A^{t}\text{.}\)
- The magnitude of transpose of a matrix is the magnitude of the matrix, i.e., \(|A^{t}| = |A|\text{.}\)
