If you pick a particular basis for

then

can be represented by a matrix

, and

can be represented by a matrix

. We are told that

, so

. Therefore,

is such a linear trasformation so that its corresponding matrix commutes with all other matrices. In order to complete your problem we need to show that

is a scalar multiple of the identity matrix. Let

. Define

the matrix so that the

entry is

and everything else is

. Notice that

is always invertible. Thus, if

commutes with every matrix it means

. If

the

-entry in that matrix equation tells us that

i.e. the matrix

has all its diagnol entries equal. Now consider the matrix equation

the

-entry tells us

. Thus,

is a scalar of a diagnol matrix.