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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

| top row: a , b  bottom row: c , d |  ... multiply the a and the d multiply down and add...

     | top row: a , b  bottom row: c , d | ... multiply the c and the b and multiply up and subtract.

 

| top row: a , b  bottom row: c , d | ... multiply the a and the d ... +          | top row: a , b  bottom row: c , d | ... multiply the c and the b ... -     ad - cb