How to Invert a Matrix

Let’s review my Matrix Inversion cheat-sheet, for those times you need to do it by hand:

For a 2x2 Matrix:

a b
c d

Then the inverse is 1/(ad-cb) * d -b
                                -c a

For a 3x3 matrix:
  a b c
  d e f
  g h i

The Inverse is 

        1                           (ei-fh)   (bi-ch)   (bf-ce)
-----------------------------   *   (fg-di)   (ai-cg)   (cd-af)
a(ei-fh) - b(di-fg) + c(dh-eg)      (dh-eg)   (bg-ah)   (ae-bd)

Inverse of a "diagonal" matrix:

a 0 0   1/a  0   0
0 b 0 =  0  1/b  0
0 0 c    0   0  1/c

This works for a diagonal matrix of any size.

Lastly, a tactic that works for any type or size of matrix:

Define the Identity Matrix:
         1 0 0
Id =     0 1 0
         0 0 1 
(etc...build an Identity Matrix to a size that matches your matrix)

Now write [A|Id]-->Do Row Operations --> Get[Id|Inv(A)]

Recall that Row Operations are actions like New Row 2 =Row 1-3*Row 2. Here’s a worked example of inverting this way: http://math.uww.edu/~mcfarlat/inverse.htm