Prove that if $A\ge B$ then $\left[ {\begin{array}{*{20}{c}} A & B \\ B & A \\ \end{array}} \right]\ge0$. Let $A$ and $B$ be $n \times n$ matrices, i.e., $A, B \in  M_n$. Also, $A \ge 0$, $B \ge 0$, and $A-B \ge 0$ which mean all these matrices are semi-positive-definite.
Why does 
$\left[ {\begin{array}{*{20}{c}}
   A & B  \\
   B & A  \\
\end{array}} \right] \ge 0$?

$\left[ {\begin{array}{*{20}{c}}
   A & B  \\
   B & A  \\
\end{array}} \right] =\left[ {\begin{array}{*{20}{c}}
   A-B & 0  \\
   0 & A-B \\
\end{array}} \right] +\left[ {\begin{array}{*{20}{c}}
   B & B  \\
   B & B  \\
\end{array}} \right] \ge 0$
