How to prove the condition of commutativity of the product of two square matrices

Question

There is a question in my textbook as follows:

Prove that for a square matrix $A$ to be commutable with any other square matrix $B$ (meaning that $AB=BA$), it is necessary and sufficient for $A$ to be a scalar matrix(i.e., to be of the form $cI$, where $c$ is a scalar and $I$ is the identity matrix).

I have proved the sufficiency part as below:

Given $A=cI$

Then $AB=cIB=cB$ and $BA=cBI=cB$.

Hence, it is proved that $AB=BA$.

But I cannot prove the necessity part, i.e., given $AB=BA$, I cannot prove that $A$ is of the form $cI$.

Please anyone help me solve it. Thanks in advance.

Answer 1

Hint to get you started:

You know that $A$ commutes with all matrices $B$. It thus also commutes with elementary matrices (matrices that are $1$ on one position and $0$ elsewhere). What can you conclude from this?