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

• Nov 25th 2010, 06:09 PM
juanma101285
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 (Shake). 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?
• Nov 25th 2010, 06:27 PM
dwsmith
$\displaystyle \displaystyle \begin{bmatrix} a & -b\\ b & a \end{bmatrix}*\begin{bmatrix} c & -d\\ d & c \end{bmatrix}=\begin{bmatrix} ac-bd & -ad-bc\\ bc+ad & -bd+ac \end{bmatrix}\rightarrow\begin{bmatrix} ac-bd & -ad-bc\\ ad+bc & ac-bd \end{bmatrix}$

$\displaystyle \displaystyle \begin{bmatrix} x & -y\\ y & x \end{bmatrix}$