Found this video showing the rooting procedure. Should be helpful for someone in doubts.
adb push Samsung_Galaxy_Tab_10.1_root.zip /sdcard/Samsung_Galaxy_Tab_10.1_root.zip
adb reboot recovery
I was under the impression that the Galaxy Tab 10.1 devices at I/O did not have an SD card. How is it being pushed to one?
What recovery are you booting into to get an option to flash the update? Not the factory one I suspect