Question 560099
<pre>
{{{(matrix(4,4,
1,1,1,1,
1,2,-1,2,
1,-1,2,1,
1,3,3, 2))}}}

Augment the matrix with the identity matrix with the 
same dimensions (4×4):

{{{(matrix(4,9,
1,1,1,1,"|",1,0,0,0,
1,2,-1,2,"|",0,1,0,0,
1,-1,2,1,"|",0,0,1,0,
1,3,3, 2,"|",0,0,0,1))}}}

We will use row operations until we get the identity on the left.

Multiply row 1 by -1 and add it to row 2:

{{{(matrix(4,9,
1,1,1,1,"|",1,0,0,0,
0,1,-2,1,"|",-1,1,0,0,
1,-1,2,1,"|",0,0,1,0,
1,3,3, 2,"|",0,0,0,1))}}}

Multiply row 1 by -1 and add it to row 3:

{{{(matrix(4,9,
1,1,1,1,"|",1,0,0,0,
0,1,-2,1,"|",-1,1,0,0,
0,-2,1,0,"|",-1,0,1,0,
1,3,3, 2,"|",0,0,0,1))}}}

Multiply row 1 by -1 and add it to row 4:

{{{(matrix(4,9,
1,1,1,1,"|",1,0,0,0,
0,1,-2,1,"|",-1,1,0,0,
0,-2,1,0,"|",-1,0,1,0,
0,2,2, 1,"|",-1,0,0,1))}}}

Multiply row 2 by -1 and add it to row 1:

{{{(matrix(4,9,
1,0,3,0,"|",2,-1,0,0,
0,1,-2,1,"|",-1,1,0,0,
0,-2,1,0,"|",-1,0,1,0,
0,2,2, 1,"|",-1,0,0,1))}}}

Multiply row 2 by 2 and add it to row 3:

{{{(matrix(4,9,
1,0,3,0,"|",2,-1,0,0,
0,1,-2,1,"|",-1,1,0,0,
0,0,-3,2,"|",-3,2,1,0,
0,2,2, 1,"|",-1,0,0,1))}}}


Multiply row 2 by -2 and add it to row 4:

{{{(matrix(4,9,
1,0,3,0,"|",2,-1,0,0,
0,1,-2,1,"|",-1,1,0,0,
0,0,-3,2,"|",-3,2,1,0,
0,0,6,-1,"|",-1,-2,0,1))}}}

Add row 3 to row 1:

{{{(matrix(4,9,
1,0,0,2,"|",-1,1,1,0,
0,1,-2,1,"|",-1,1,0,0,
0,0,-3,2,"|",-3,2,1,0,
0,0,6,-1,"|",-1,-2,0,1))}}}

Multiply row 3 by 2 and add to row 4:

{{{(matrix(4,9,
1,0,0,2,"|",-1,1,1,0,
0,1,-2,1,"|",-1,1,0,0,
0,0,-3,2,"|",-3,2,1,0,
0,0,0,3,"|",-5,2,2,1))}}}

To avoid fractions multiply row 3 by -2 and row 2 by 3:

{{{(matrix(4,9,
1,0,0,2,"|",-1,1,1,0,
0,3,-6,3,"|",-3,3,0,0,
0,0,6,-4,"|",6,-4,-2,0,
0,0,0,3,"|",-5,2,2,1))}}}

Add row 3 to row 2:

{{{(matrix(4,9,
1,0,0,2,"|",-1,1,1,0,
0,3,0,-1,"|",3,-1,-2,0,
0,0,6,-4,"|",6,-4,-2,0,
0,0,0,3,"|",-5,2,2,1))}}}

Again to avoid fractions multiply row 4 by 2 and row 1 by -3:

{{{(matrix(4,9,
-3,0,0,-6,"|",3,-3,-3,0,
0,3,0,-1,"|",3,-1,-2,0,
0,0,6,-4,"|",6,-4,-2,0,
0,0,0,6,"|",-10,4,4,2))}}}
 
Add row 4 to row 1:

{{{(matrix(4,9,
-3,0,0,0,"|",-7,1,1,2,
0,3,0,-1,"|",3,-1,-2,0,
0,0,6,-4,"|",6,-4,-2,0,
0,0,0,6,"|",-10,4,4,2))}}}

To avoid fractions multiply row 2 by 6:

{{{(matrix(4,9,
-3,0,0,0,"|",-7,1,1,2,
0,18,0,-6,"|",18,-6,-12,0,
0,0,6,-4,"|",6,-4,-2,0,
0,0,0,6,"|",-10,4,4,2))}}}

Add row 4 to row 2:

{{{(matrix(4,9,
-3,0,0,0,"|",-7,1,1,2,
0,18,0,0,"|",8,-1,-8,2,
0,0,6,-4,"|",6,-4,-2,0,
0,0,0,6,"|",-10,4,4,2))}}}


To avoid fractions multiply row 4 by 2 and row 3 by 3:

{{{(matrix(4,9,
-3,0,0,0,"|",-7,1,1,2,
0,18,0,0,"|",8,-1,-8,2,
0,0,18,-12,"|",18,-12,-6,0,
0,0,0,12,"|",-20,8,8,4))}}}

Add row 4 to row 3:

{{{(matrix(4,9,
-3,0,0,0,"|",-7,1,1,2,
0,18,0,0,"|",8,-1,-8,2,
0,0,18,0,"|",-2,-4,2,4,
0,0,0,12,"|",-20,8,8,4))}}}

Now get 1's on the diagonal:

Divide row 1 by -3
Divide row 2 by 18
Divide row 3 by 18
Divide row 4 by 12

{{{(matrix(4,9,
1,0,0,0,"|",7/3,-1/3,-1/3,-2/3,
0,1,0,0,"|",4/9,-1/9,-4/9,1/9,
0,0,1,0,"|",-1/9,-2/9,1/9,2/9,
0,0,0,1,"|",-5/3,2/3,2/3,1/3))}}}

Now that we have the identity on the left, the 
augmented part is the inverse:

{{{(matrix(4,4,
7/3,-1/3,-1/3,-2/3,
4/9,-1/9,-4/9,1/9,
-1/9,-2/9,1/9,2/9,
-5/3,2/3,2/3,1/3))}}}

Edwin</pre>