This is a trick that ONLY works for 3 x 3's. You cannot use it for 4 x 4's and higher... For these, the formal approach is a gnarly thing that expands around a row or column and uses critters called "minors." However, I highly recommend a computer or graphing calculator. If you go on to take Calculus 3 or Linear Algebra, you'll need the "real" 3 x 3 technique, but it will be specialized and really easy to pick up.
On to the trick...
It works a lot like the 2 x 2 method where you
|multiply down and add...|
|and multiply up and subtract.|