Example-1 :-Solution :-
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.Solution :-
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.Solution :-
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.