Consider the matrix M = 3 -1
4 -1
Verify that M^2= 2M -I
the question is what is I ??
Generally, when talking about matrices, I is the identity element. ie. the element such that for any matrix A, A*I = A.
It's the integer equivalent of 1 under multiplication.
ie. if A is an integer A*1 = A.
For 2x2 matrices the identity has
[ 1 0 ]
[ 0 1 ]
for 3x3s it's
[1 0 0 ]
[0 1 0 ]
[0 0 1 ]
And so forth. I don't know how to draw matrices in this thing... sorry...
I assume that you can verify the given result. Then:
$\displaystyle M^2 = 2M - I \Rightarrow M^{-1} M^2 = 2 M^{-1} M - M^{-1}$ $\displaystyle \Rightarrow M = 2I - M^{-1} \Rightarrow M^{-1} = 2I - M = \, .... $.
Side note: $\displaystyle M^2 = 2M - I \Rightarrow M^2 - 2M + I = 0$ has been got using the Cayley-Theorem Theorem: Cayley-Hamilton Theorem
Edit: Whoops, I just realised that your question wasn't find the inverse of M. Not to worry, since you asked what I is I figure I've probably anticipated what your next question is.