Question 190355
Simply append the 3x3 identity matrix {{{(matrix(3,3,1,0,0,0,1,0,0,0,1))}}} to the given matrix to get


  
 {{{(matrix(3,6,1,1,1,1,0,0,2,1,1,0,1,0,2,2,3,0,0,1))}}}



Now use Gauss-Jordan Elimination (ie row reduce) to transform the left hand block matrix to the 3x3 identity matrix {{{(matrix(3,3,1,0,0,0,1,0,0,0,1))}}}. The right hand block 3x3 matrix will be the inverse of the given matrix.



So here are the steps needed to row reduce (provided by the <a href="http://www.math.odu.edu/~bogacki/lat/">Linear Algebra Toolkit</a>):


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/rref-1.png">



Notice how the left hand block matrix is  the 3x3 identity matrix {{{(matrix(3,3,1,0,0,0,1,0,0,0,1))}}}, so this means that 1) the inverse of A exists (and is unique), and 2) the right hand matrix is the inverse of A



Since the right hand block 3x3 matrix is {{{(matrix(3,3,-1,1,0,4,-1,-1,-2,0,1))}}}, this means that if {{{A=(matrix(3,3,1,1,1,2,1,1,2,2,3))}}}, then {{{A^(-1)=(matrix(3,3,-1,1,0,4,-1,-1,-2,0,1))}}}