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## Homework Statement

Find all 2x2 matrices X such that AX=XA for all 2x2 matrices

## The Attempt at a Solution

Let A =

a b

c d

and X =

w x

y z

Then AX = XA ==>

aw+by=wa+xc ...(1)

ax+bz=wb+xd ...(2)

cw+dy=ya+zc ...(3)

cx+dz=yb+zd ...(4)

(1) ==> by = xc, which holds for all b and c only if x=y=0.

(2) ==> bz = wb, which hods for all b only if z=w.

(3) ==> w=z

(4) ==> x=y=0

So the answer is x=y=0, and w=z. That is, X = k*I with k being any real number and I is the identity matrix.

How does this look?