# Transpose matrix   - properties and formulas -

- Definition
- Examples
Definition
A matrix in which the row and column elements of a matrix are exchanged is called the transposed matrix of that matrix. The transpose of a matrix $A$ is represented as $A^ {T}$. The $i$-th row, j-th column element of $A^T$ is the $j$-th row, $i$-th column element of $A:$

Explanation
The $i$-th row, $j$-th column element of $A^T$ is the $j$-th row, $i$-th column element of $A$. For example, if $i=1$ and $j=2$,
And if $i=2$ and $j=1$,
By transposition, each element is exchanged corssing diagonal elements.
Since $(A^{T})_{ii} = A_{ii}$, each diagonal element is unchanged.
In general, the size of a traspose matrix is different from its original one. If $A$ is an $m \times n$ matrix, then $A^{T}$ is an $n \times m$ matrix.
If $A$ is a square matrix, the size of $A^{T}$ is the same as that of $A$.
A column matrix has only one column but any number of rows. A row matrix has only one row but any number of columns. Transpose transforms a column matrix to a row matrix, and vice versa.
Various matrices are defined by transpose. For example,
• If $A^{T}=A$, $A$ is called a symmetric matrix.
• If $A^{T}A=AA^{T}=I$, $A$ is called an orthogonal matrix.