Transposed matrix/Properties/Exercise

Let ${\displaystyle {}K}$ be a field, and ${\displaystyle {}m,n,p\in \mathbb {N} }$. Prove that the transpose of a matrix satisfies the following properties (where ${\displaystyle {}A,B\in \operatorname {Mat} _{m\times n}(K)}$, ${\displaystyle {}C\in \operatorname {Mat} _{n\times p}(K)}$, and ${\displaystyle {}s\in K}$).

1. ${\displaystyle {}{({A^{\text{tr}}})^{\text{tr}}}=A\,.}$

2. ${\displaystyle {}{(A+B)^{\text{tr}}}={A^{\text{tr}}}+{B^{\text{tr}}}\,.}$

3. ${\displaystyle {}{(sA)^{\text{tr}}}=s\cdot {A^{\text{tr}}}\,.}$

4. ${\displaystyle {}{(A\circ C)^{\text{tr}}}={C^{\text{tr}}}\circ {A^{\text{tr}}}\,.}$