The exercise says to find all the matrices that commute with

[ 1 0]

[-1 0]

I do

[ 1 0] _ [a b] ___ [a b] ___ [a b] _ [ 1 0] __ [a-b 0]

[-1 0] x [c d] = [-a -b] and [c d] x [-1 0] = [c-d 0].

(I inserted some underscores so the above reads better. The underscores don't denote anything)

Since the two resulting matrices must be equal I get a=a-b, b=0, -a=c-d and -b=0.

At this point I get stuck so I look at the solution which says "for equality it is necessary to have a=a-b, b=0, -a=c-d. thus those matrices that commute with the given matrix are all matrices of the form

[a 0]

[c c+a] where a and c can take any real values."

I don't understand this. How do you get from "a=a-b, b=0, -a=c-d" to

[a 0]

[c c+a] ?

Any ideas? The book (Linear Algebra: With Applications - Gareth Williams - Google Books) doesn't show how to find commuting matrices. I couldn't find any such instructions at the web.

Thanks