Question

E and F are symmetric matrices of dimension n. (Note that symmetric matrices must be square!) Show that E'FE is symmetric, but that EF need not be symmetric.

Answer 1

Answer with Step-by-step explanation:

We are given that E and F are symmetric matrices of dimension n.

Therefor, E'=E and F'=F

We are given that EF need not be symmetric

If EF may not  be commutative then EF necessarily may  not be symmetric.

We have to show that E'FE is symmetric .

It means we have to show that (E'FE)'=E'FE

$(E'FE)'=(E'(FE))'$

$(E'FE)'=(FE)'(E')'$

We know that (A')'=A and (AB)'=B'A'

Using the property

$(E'FE)'=E'F'E$

$(E'FE)'=E'FE$ (F'=F)

Hence, proved.