There’s a newer, easier method to root your Galaxy Tab 10.1 instead the old method.
This one is pretty simple.
First, you will need Windows computer. If you don’t have one, borrow a Windows laptop from your friend or family member.
Second, make sure you’ve installed Galaxy Tab 10.1 Windows drivers. (Download it here)
Third, put your Galaxy Tab 10.1 into ODIN download mode by holding down both Power Button and Volume Down buttons. Right after the Samsung logo shows, you will see two menus, let go of both buttons immediately and then hit Volume Up button, you should be in ODIN download mode.
Fourth, download the tabrooteasy.zip file and unzip:
Run the Odin3 v1.85.exe program, then choose PDA, choose “recovery-cwm_18.104.22.168-sam-tab-10.1.tar.md5″ and hit Start.
You can reboot your Tab 10.1 by holding down the Power button.
Copy over the file “Samsung_Galaxy_Tab_10.1_root.zip” from the tabrooteasy folder to the root folder of your Tab 10.1.
Reboot into ClockworkMod Recovery by holding down both Power Button and Volume Down buttons. Right after the Samsung logo shows, you will see two menus, let go of both buttons immediately and then hit Volume Down, then Volume Up button, you should be in ClockworkMod Recovery.
Once you are in ClockworkMod Recovery, choose “install zip from sdcard” (using Volume buttons) and hit the Power button.
Next, choose “choose zip from sdcard” and hit the Power button.
We are going to zip the Root/SU files again by going to “choose zip from sdcard” and hit the Power button.
Choose the Samsung_Galaxy_Tab_10.1_root.zip file and hit the Power button, the hit the “Yes” button.
When that is done, you just need to reboot your Tab 10.1 and you should have a fully-rooted Android 3.1 Galaxy Tab 10.1.