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Math Help - Showing that the product of 2 matrices in a set of matrices is also in the set

  1. #1
    Junior Member
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    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?
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  2. #2
    MHF Contributor
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    Florida
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    Thanks
    5
    \displaystyle \begin{bmatrix}<br />
a & -b\\ <br />
b & a<br />
\end{bmatrix}*\begin{bmatrix}<br />
c & -d\\ <br />
d & c<br />
\end{bmatrix}=\begin{bmatrix}<br />
ac-bd & -ad-bc\\ <br />
bc+ad & -bd+ac<br />
\end{bmatrix}\rightarrow\begin{bmatrix}<br />
ac-bd & -ad-bc\\ <br />
ad+bc & ac-bd<br />
\end{bmatrix}

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

    \displaystyle \begin{bmatrix}<br />
x & -y\\ <br />
y & x<br />
\end{bmatrix}
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  3. #3
    Junior Member
    Joined
    Sep 2010
    Posts
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    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.
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