Originally Posted by

**registered0** Hi everyone, I've been stuck on this question:

A =

a b

c c

B =

x y

z w

Show that AB = BA for any A only if B is a multiple of the identity matrix.

---

AB =

(ax+bz) (ay+bw)

(cx+dz) (cy+dw)

BA =

(xa+yc) (xb+yd)

(za+wc) (zb+wd)

equating elements gives

ax + bz = xa + yc

ay + bw = xb + yd

cx + dz = za + wc

cy + dw = zb + wd

cancel out common terms

bz = yc

ay + bw = xb + yd

cx + dz = za + wc

cy = zb, same as first equation

and I'm stuck here.

bz = yc

ay + bw = xb + yd

cx + dz = za + wc

For these things to be equal, we would need

a = d, w = x, and bz = yc

but I don't know what to do next.