Question

If a is a diagonal matrix with all its diagonal elements different ,prove that ab=ba

Answer 1

fixed Proof (Confirm please):

Let A,B be two diagonal matrices of order n. Then, both AB,BA are defined and are of the same order n (i.e. sizes match). Also, Aij,Bij=0 whenever i≠j.

Consider the case i≠j:

(AB)ij=∑k=1nAikBkj=∑i≠k=jAikBkj+∑i=k≠jAikBkj+∑i≠k≠jAikBkj=∑i≠k=j0⋅Bkj+∑i=k≠jAik⋅0+∑i≠k≠j0=0+0+0=0=∑i=k≠jBik⋅0+∑i≠k=j0⋅Akj+∑i≠k≠j0=∑i=k≠jBik⋅Akj+∑i≠k=jBik⋅Akj+∑i≠k≠jAikBkj=∑k=1nBikAkj=(BA)ij

Consider the case i=j

(AB)ij=(AB)ii=∑k=1nAikBki=∑k≠iAikBki+∑k=iAikBki=∑k≠i0⋅0+AiiBii=0+AiiBii=0+BiiAii=∑k≠i0⋅0+BiiAii=∑k≠iBikAki+∑k=iBikAki=∑k=1nBikAki=(BA)ii=(BA)ij