You can use putSerializable. Arrays are serializable.
To store:
bundle.putSerializable("list", selected_list);
// Here bundle is Bundle object.
To access:
String[][] passedString_list = (String[][]) bundle.getSerializable("list");
Example
Intent mIntent = new Intent(this, Example.class);
Bundle mBundle = new Bundle();
mBundle.putSerializable("list", selected_list);
mIntent.putExtras(mBundle);