THE INVERSE OF AN n x n MATRIX
See our text ( Rolf, Page 163) for a discussion of matrix inverses. Briefly, matrix
inverses behave as reciprocals do for real numbers : the product of a matrix
and it's inverse is an identity matrix. 
Let A be the name of our nxn matrix: nonsquare matrices have no inverse.
The following steps will produce the inverse of A, written A^{1}. Note the similarity
between this method and GAUSS/JORDAN method, used to solve a system of
equations.
[2] 
Pivot on matrix
elements in positions 11, 22, 33, continuing through
nn in that order, with the goal of creating a copy
of the identity matrix
I_{n} in the left
portion of the augmented matrix. 
If one of the pivoting elements is zero, then first interchange
it's row with a lower row. If no such interchange produces
a nonzero pivot element, then the matrix A has no inverse. This
step [2] is equivalent to step 2 on Pg 163 of our text Rolf,
resulting in (REDUCED) DIAGONAL FORM.
See an example below, and try the
Pivot Engine when you check your
pivoting skills. 
[ A  I_{n} ]
===> [ I_{n}
 A^{1} ] 


[3] 
When step [2] above is done, the right half of the latest
augmented matrix will be the desired inverse,
A^{1}; write it separately, and you're done,
as in the example below. 

Note 1 :

Professor M^{c}Farland names
row operations just a bit
differently from our text: follow Prof McFarland's naming style. 
Note 2 : Check out Prof M^{c}Farland's
interactivePIVOT ENGINE
as you use row operations.
Note 3 : Compare the above 3 steps for
those used in GAUSS/JORDAN.
EXAMPLE OF FINDING THE INVERSE OF A MATRIX A
See our text (Rolf, Pg 163) for one example; below is another example :
We must find the inverse of the matrix A at the right 

A = 

1 2 2 
1 1 2 
3 2 1 




Below is the same matrix A, augmented by
the 3x3 identity
matrix. The first pivot encicled in red 

Below are the row operations required for the first
pivoting 
Next pivot on "3" in the 22 position below, encircled in red 

The columns of the 3x3 identity matrix are colored blue
as they reappear on the left side 


Below is the result of performing
P1, so the pivot
(22 position) is now "1". Next we perform
P2 

Row operations
of P2
are below 
The result of the second pivoting is below. We now
pivot on the
element in the 33 position, encircled in red below 


Below is the result of performing P1, so
the pivot (33 position) is now "1". Next we perform
P2. 

Below are the row operations of P2 
The result of the third (and last) pivoting is below with
3x3 identity matrix in blue 

The matrix below is NOT A^{1} 



(REDUCED) DIAGONAL FORM
E 

Thus, our final step is to
separate the desired inverse
from the above matrix: 

A^{1} = 



Note : THE MATRIX INVERSE METHOD for solving a system of equations will use
the above discussion, and even continue the above problem.