Nonzero Matrcies A; B; C such that AC = BC and A does not equal B?

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Homework Help Overview

The discussion revolves around finding nonzero matrices A, B, and C such that the equation AC = BC holds true while ensuring that A is not equal to B. The subject area pertains to linear algebra and matrix theory.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants express uncertainty about how to begin solving the problem and seek guidance on completing it. Some suggest exploring simple non-zero matrices and manipulating them. Others analyze the implications of the equation AC = BC, leading to the expression (A-B)C = 0, prompting questions about the existence of non-zero matrices that can multiply to zero.

Discussion Status

The discussion is ongoing, with participants exploring various approaches and questioning assumptions related to matrix multiplication. Some have suggested looking into singular matrices, indicating a potential direction for further exploration.

Contextual Notes

Participants have noted the absence of specific equations or methods to guide their attempts, highlighting the exploratory nature of the problem.

sheldonrocks97
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Homework Statement



Find nonzero matrices A; B; C such that AC = BC
and A does not equal B

Homework Equations



None that I know about.

The Attempt at a Solution



I'm not sure where to start, I would like to know how to complete this problem.
 
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sheldonrocks97 said:

Homework Statement



Find nonzero matrices A; B; C such that AC = BC
and A does not equal B

Homework Equations



None that I know about.

The Attempt at a Solution



I'm not sure where to start, I would like to know how to complete this problem.

The simplest non-zero matrices are
[tex] \begin{pmatrix}<br /> 1 & 0 \\ 0 & 0 <br /> \end{pmatrix},\quad<br /> \begin{pmatrix}<br /> 0 & 1 \\ 0 & 0 <br /> \end{pmatrix},\quad<br /> \begin{pmatrix}<br /> 0 & 0 \\ 1 & 0 <br /> \end{pmatrix},\quad<br /> \begin{pmatrix}<br /> 0 & 0 \\ 0 & 1<br /> \end{pmatrix}[/tex]
Play around.
 
If AC = BC, then AC - BC = 0. I.e. (A-B)C = 0.

Can you find two non-zero matrices that multiply to zero?
 
Find out about singular matrices.
 

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