Does Commutativity Hold for Matrices A and B with a Specific Matrix C?

[TheScienceAlliance]

If A and B are matrices that AC = AC and BC=CB, where C is a matrix whose first row's entries are 0 1 and the second row's entries are -1 0, then AB=BA.

 

[topsquark]

If A and B are matrices that AC = AC and BC=CB, where C is a matrix whose first row's entries are 0 1 and the second row's entries are -1 0, then AB=BA.

Is there a typo? Did you mean AC = CA?

-Dan

 

[TheScienceAlliance]

Is there a typo? Did you mean AC = CA?

-Dan

Yes. I apologize.

 

[topsquark]

Okay, so this is more or less a construction proof. You know that
\(\displaystyle C = \left ( \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right )\)

Let
\(\displaystyle A = \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right )\)

and
\(\displaystyle B = \left ( \begin{matrix} w & x \\ y & z \end{matrix} \right )\)

So.
1) Using AC = CA show that
\(\displaystyle A = \left ( \begin{matrix} a & b \\ -b & a \end{matrix} \right )\)

2) Using BC = CB show that
\(\displaystyle B = \left ( \begin{matrix} w & x \\ -x & w \end{matrix} \right )\)

3) Using A and B from 1) and 2) show that AB = BA.

-Dan