Question

if a diagonal matrix is commutative with every matrix of the same order,then it is necessarily a ______ matrix
a. zero matrix
b. unit matrix
c. scalar matrix
d. a symmetric matrix
the ans is c... please explain how?... @Mathematics

Answer 1

Consider a general diagonal matrix to be A=[a 0,0,b] and other matrix B=[w x , y z] then compute AB then BA and then equate both which will you a=b which is a scalar matrix ....

Answer 2

Multiplication with an identity matrix is commutative. As every scalar matrix can be written as a product of a scalar and the Identity matrix, most appropriate answer is (c) part.

PS: The property will be satisfied by option (a) and (b) also.
Every diagonal matrix is a symmetric matrix.