Cayley-Hamilton Tricks

Diary, OMG!

There's a super slick trick to find the inverse of a matrix.  Consider a 2 x 2 matrix and write out the characteristic polynomial.  For example, consider a matrix A

column 1:  2  1
column 2:  0  1

with characteristic polynomial in x:

1-2x+x^2

But the Cayley Hamilton theorem says that we can replace the variable x by the matrix A.  So,

1-2A+A^2=0

And multiplying on the left by the inverse gives

A^-1=2-A

WHICH IS MIND BLOWING!  Where has this trick been all my mathematical life?!  It's on the same short cutting plane as Gaussian Elimination to solve systems of equations or finding out that Leif Ericson actually discovered America before Columbus!