STEP 3:

Remember that our goal is the make the left side the identity matrix...

    [ row 1: 2 , 0  row 2: 0 , 1  |  row 1: -5 , -3  row 2: 2 , 1 ] ... make the 2 and the 1 in the left half one's

                                                   This guy is already OK.

 

[ row 1: 1 , 0  row 2: 0 , 1  |  row 1: -( 5 / 2 ) , -( 3 / 2 )  row 2: 2 , 1 ] ... ( 1 / 2 ) times Row 1 ... believe it or not, the right half should be A^( -1 )

 

A^( -1 ) = [ row 1: -( 5 / 2 ) , -( 3 / 2 )  row 2: 2 , 1 ]

So... how do we know for sure?

Well, think back to regular numbers for a minute...

 What's( 1 / 3 ) times 3? 
 

 

( 1 / 3 ) times 3 = 1the multiplicative inverse

With letters...

( 1 / a ) times a = 1

Soa^( -1 ) times a = 1