$A_1^T=-A_1, A_2^T=-A_2, \dots .., A_{20}^T=-A_{20}$

$B=\sum _{r=1}^{20}2r{\left(A_r\right)}^{2r+1}$
$$\begin{gathered} B^T={\left(\sum _{r=1}^{20}2r{\left(A_r\right)}^{2r+1}\right)}^T \\ =\sum _{r=1}^{20}2r{\left(A_r^T\right)}^{2r+1} \end{gathered}$$
$$\begin{gathered} =\sum _{r=1}^{20}2r{\left(-A_r\right)}^{2r+1} \\ =-\sum _{r=1}^{20}2r{\left(A_r\right)}^{2r+1} \end{gathered}$$

$=-B$
$\Rightarrow B$ is skew-symmetric

Hence, the sum of principal diagonal elements of $B=0$