Problem:

Determine the subspace of R(2 by 2) consisting of all matrices that commute with the given matrix:

(1 0 ) =B

(0 -1)

Attempt at sol'n:

I wanted a general 2 by 2 matrix A so that AB=BA. Multiplying out component wise, I think that it must be the case that a11=a11, a12=-a12=0, -a21=a21=0, and -a22=-a22. So I think that

(1 0 ) (a11 0 ) = (a11 0 ) (1 0 )

(0 -1) (0 -a22) (0 -a22) (0 -1)

And I already know that if S is a subspace of R(n by n) where S is the set of all matrices that commute with a fixed vector in R(n by n) that S is a subspace of R(n by n).

So is the above sufficient or do I need to do something entirely different?