Operation of Transpose of a Matrix and its Properties

The transpose of $A=[a_{ij}]_{m\times n}$, written $A^T$, is the $n\times m$ matrix obtained by turning every row of $A$ into the corresponding column (equivalently, every column into the corresponding row): $A^T=[b_{ij}]_{n\times m}$ where $b_{ij}=a_{ji}$, so the $(i,j)$th entry of $A^T$ is the $(j,i)$th entry of $A$.

Basic properties (for matrices $A,B$ of suitable order, $k$ any scalar):

1. $(A^T)^T=A$ — transposing twice restores the original matrix.
2. $(kA)^T=kA^T$.
3. $(A+B)^T=A^T+B^T$.
4. $(AB)^T=B^T\,A^T$ — the reversal law: the transpose of a product equals the product of the transposes in reverse order, not $A^TB^T$. …