Example-1 :- 

Example-2 :-  If A and B are symmetric matrices of the same order, then show that AB is symmetric if and only if A and B commute, that is AB = BA.

  Since A and B are both symmetric matrices, therefore A′ = A and B′ = B.
  Let AB be symmetric, then (AB)′ =AB 
  But (AB)′ =B ′A′= BA
  Therefore BA = AB Conversely, if AB = BA, then we shall show that AB is symmetric. 
  Now (AB)′ =B ′A′ = BA (as A and B are symmetric) = AB 
  Hence AB is symmetric.

Example-3 :-  Find a matrix D such that CD – AB = O.

  Since A, B, C are all square matrices of order 2, and CD – AB is well defined, D must be a square matrix of order 2.