# Matrix Transpose

Let $A∈M_{m×n}(R)$. The **transpose** of $A$, denoted by $A_{T}$, is the $n×m$ matrix satisfying $(A_{T})_{ij}=(A)_{ji}$.

**Properties of Transpose**. Let $A,B∈M_{m×n}(R)$ and $c∈R$. Then
$A_{T}∈M_{m×n}(R)$
$(A_{T})_{T}=A$
$(A+B)_{T}=A_{T}+B_{T}$
$(cA)_{T}=cA_{T}$

A matrix $A$ is *symmetric* if $A_{T}=A$.

Other property: $(AB)_{T}=B_{T}A_{T}$