Bi invariant metrics on $SL_n(\mathbb{R})$ Does there exist a bi-Invariant metric on $SL_n(\mathbb{R})$. I tried to google a bit but I didn't find anything helpful. 

If $d$ were a bi-invariant metric on $\operatorname{SL}_n(\mathbb{R})$, we would be able to restrict it to a bi-invariant metric on $\operatorname{SL}_2(\mathbb{R})$ as in Jason's answer.
And then we would have $$ d\left(\left[{\begin{array}{cc}
   t & 0 \\
   0 & 1/t \\
  \end{array}}\right]\left[{\begin{array}{cc}
   1 & 1 \\
   0 & 1 \\
  \end{array}}\right]\left[{\begin{array}{cc}
   1/t & 0 \\
   0 & t \\
  \end{array}}\right], \left[{\begin{array}{cc}
   1 & 0 \\
   0 & 1 \\
  \end{array}}\right]\right)= d\left(\left[{\begin{array}{cc}
   1 & t^2 \\
   0 & 1 \\
  \end{array}}\right], \left[{\begin{array}{cc}
   1 & 0 \\
   0 & 1 \\
  \end{array}}\right]\right)\to 0$$
as $t\to 0$. On the other hand, we also have 
$$
d\left(\left[{\begin{array}{cc}
   t & 0 \\
   0 & 1/t \\
  \end{array}}\right]\left[{\begin{array}{cc}
   1 & 1 \\
   0 & 1 \\
  \end{array}}\right]\left[{\begin{array}{cc}
   1/t & 0 \\
   0 & t \\
  \end{array}}\right], \left[{\begin{array}{cc}
   1 & 0 \\
   0 & 1 \\
  \end{array}}\right]\right)= d\left( \left[{\begin{array}{cc}
   1 & 1 \\
   0 & 1 \\
  \end{array}}\right], \left[{\begin{array}{cc}
   1 & 0 \\
   0 & 1 \\
  \end{array}}\right]\right) \neq 0.
$$
