See the attached screenshot for the question - too complex to type out.

Exercise 4. Let A be an n xn matrix. In this problem, we will show that if AB = BA for all BE Mn (R), then A = r . In for some r E R is a scalar matrix. (1) First check that if A = r . In, then AB = r . B = BA for all BE Mn(R). (2) Let A = (aij)nxn. Denote by Eij the elementary n x n matrix whose ij-th entry is equal to 1 and all other entries are zero. Compute AEij and Eij A. (3) Suppose AEij = EijA for all 1 _ , j _ n. Show that A is a scalar matrix by checking . all = a22 = . . . = ann. . aj = 0 when i * j