Thread: Showing that the product of 2 matrices in a set of matrices is also in the set

Hi, I have managed to show that the sum of 2 matrices in one set is still in the same set, but I don't know how to show it when it is a multiplication . Thanks!

So, here is the problem:

Consider the set C of all matrices (with real entries) of the form

(sorry, I don't know how to code matrices! I'll separate each element with "|")

a | -b
b | a

Show that the product of two matrices in C is also in C.

So yeah, I have got up to here, but I don't know how to show that these is in the set C of matrices.

Let matrix M=
a | -b
b | a

So, M*M=
a^2-b^2 | -2ab
2ab | a^2-b^2

So this is what confuses me, it does not look like it was in the set?

If you want to visualize it better, let x=ac-bd and y=ad+bc; thus, -y=-(ad+bc)=-ad-bc.

Thanks so much! I actually kind of got the other part of the exercise wrong (the one of showing the sum), but I now see what I was doing wrong.