Question 255959
<pre><font size = 4 color="indigo"><b> 
{{{A=(matrix(4,4,
 -4, 0, 0, 0, 
-1, 2, 0, 0, 
-4, 4, 4, 0, 
-1, -9, -1, 1))}}}

Augment it with the identity matrix:

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

))}}}

The idea is to make the left side become the
identity matrix, and then the right side
will become the inverse:

Multiply Row 2 by -4:

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
4, -8, 0, 0,         "|", 0,-4,0,0,
-4, 4, 4, 0,         "|", 0,0,1,0, 
-1, -9, -1, 1,       "|", 0,0,0,1

))}}}

Add Row 1 to Row 2

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -8, 0, 0,         "|", 1,-4,0,0,
-4, 4, 4, 0,         "|", 0,0,1,0, 
-1, -9, -1, 1,       "|", 0,0,0,1

))}}}

Multiply Row 3 by -1

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -8, 0, 0,         "|", 1,-4,0,0,
4, -4, -4, 0,         "|", 0,0,-1,0, 
-1, -9, -1, 1,       "|", 0,0,0,1

))}}}

Add Row 1 to Row 3:

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -8, 0, 0,         "|", 1,-4,0,0,
0, -4, -4, 0,         "|", 1,0,-1,0, 
-1, -9, -1, 1,       "|", 0,0,0,1  ))}}}

Multiply Row 4 by -4

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -8, 0, 0,         "|", 1,-4,0,0,
0, -4, -4, 0,         "|", 1,0,-1,0, 
4, 36, 4, -4,       "|", 0,0,0,-4  ))}}}

Add Row 1 to Row 4


{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -8, 0, 0,         "|", 1,-4,0,0,
0, -4, -4, 0,         "|", 1,0,-1,0, 
0, 36, 4, -4,       "|", 1,0,0,-4  ))}}}

Multiply  Row 3 by -2

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -8, 0, 0,         "|", 1,-4,0,0,
0, 8, 8, 0,         "|", -2,0,2,0, 
0, 36, 4, -4,       "|", 1,0,0,-4  ))}}}

Add Row 2 to Row 3:

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -8, 0, 0,         "|", 1,-4,0,0,
0, 0, 8, 0,         "|", -1,-4,2,0, 
0, 36, 4, -4,       "|", 1,0,0,-4  ))}}}

Multiply Row 2 by 9, and multiply Row 4 by 2

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -72, 0, 0,         "|", 9,-36,0,0,
0, 0, 8, 0,         "|", -1,-4,2,0, 
0, 72, 8, -8,       "|", 2,0,0,-8  ))}}}

Add Row 2 to Row 4:

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -72, 0, 0,         "|", 9,-36,0,0,
0, 0, 8, 0,         "|", -1,-4,2,0, 
0, 0, 8, -8,       "|", 11,-36,0,-8  ))}}}

Multiply Row 3 by -1:

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -72, 0, 0,         "|", 9,-36,0,0,
0, 0, -8, 0,         "|", 1,4,-2,0, 
0, 0, 8, -8,       "|", 11,-36,0,-8  ))}}}

Add Row 3 to Row 4

{{{(matrix(4,9,
 -4, 0, 0, 0,        "|", 1,0,0,0, 
0, -72, 0, 0,         "|", 9,-36,0,0,
0, 0, -8, 0,         "|", 1,4,-2,0, 
0, 0, 0, -8,       "|", 12,-32,-2,-8  ))}}}

Now divide the Row 1 through by -4,
divide Row 2 through by -72,
divide Row 3 through by -8
divide Row 4 through by -8

{{{(matrix(4,9,
 1, 0, 0, 0,        "|", -1/4,0,0,0, 
0, 1, 0, 0,         "|", -1/8,1/2,0,0,
0, 0, 1, 0,         "|", -1/8,-1/2,1/4,0, 
0, 0, 0, 1,       "|", -3/2,4,1/4,1  ))}}}

Now we have the identity on the left, so the inverse is 
the 4x4 matrix on the right, that is:

{{{A^(-1)=(matrix(4,4,
-1/4,0,0,0, 
-1/8,1/2,0,0,
-1/8,-1/2,1/4,0, 
-3/2,4,1/4,1  ))}}}

Edwin</pre>