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

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Discussion Overview

The discussion revolves around the commutativity of matrices A and B under specific conditions involving a matrix C. Participants explore whether the relationship AB = BA holds true given the constraints AC = CA and BC = CB, with C defined as a specific 2x2 matrix.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant asserts that if AC = CA and BC = CB for a specific matrix C, then AB = BA.
  • Another participant questions whether there was a typo in the original post, suggesting that AC should be CA instead.
  • A later reply confirms the typo and acknowledges the correction.
  • Further, a participant outlines a construction proof approach, proposing specific forms for matrices A and B based on the conditions provided.
  • The construction proof includes steps to derive A and B and subsequently show that AB = BA.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the initial claim regarding commutativity, as the discussion includes corrections and proposed proofs without definitive agreement on the outcome.

Contextual Notes

The discussion relies on specific definitions and assumptions about the matrices involved, and the steps to demonstrate commutativity are not fully resolved.

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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.
 
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MathHelpBoardsUser said:
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
 
topsquark said:
Is there a typo? Did you mean AC = CA?

-Dan
Yes. I apologize.
 
Okay, so this is more or less a construction proof. You know that
[math]C = \left ( \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right )[/math]

Let
[math]A = \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right )[/math]

and
[math]B = \left ( \begin{matrix} w & x \\ y & z \end{matrix} \right )[/math]

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

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

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

-Dan
 

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