The exercise says to find all the matrices that commute with
[ 1 0]
[ 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
[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
[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.