Let A & B be matrices such that:
A.B = B.A (with det(B) =/= 0)
Show that B can be written under the form:
B = mA + pI where m & p are real numbers and I is the identity matrix.
You need to make the observation that A commutes only with itself (that is, with A) and I. Once this has been noted, it clearly follows that if A commutes with B then B must be a linear combination of the matrices that commute with A, that is, a linear combination of the matrices A and I.