Finding System Solutions of the system Ax=0; A being a matrix

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Homework Help Overview

The discussion revolves around finding all solutions to the system Ax=0, where A is a specified 3x3 matrix. Participants are exploring the implications of the matrix's properties and the methods for solving the associated linear equations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to solve the system by expressing "x" as a 3x1 matrix and setting up the corresponding equations. Some participants suggest using row operations to simplify the matrix instead of solving the equations directly.

Discussion Status

Participants are actively engaging with the problem, with some providing guidance on methods such as row operations. There is a recognition of the matrix's invertibility, which leads to a discussion about the implications for the solution set.

Contextual Notes

There is uncertainty regarding the application of row operations, as the original poster mentions having learned about them recently. The discussion also reflects on the nature of the solutions to the system, particularly in the context of the matrix being invertible.

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Homework Statement



Find all solutions of the system Ax=0
where A = the 3x3 matrix

[1 3 2]
[2 6 9]
[2 8 8]

Homework Equations



not really sure what equations to include

The Attempt at a Solution



wasnt positive how to go about answering this question because I am not sure what its asking for...

i turned "x" into a 3x1 matrix consisting of x1, x2 and x3 and tried to solve the system of equations...

1x1 + 3x2 + 2x3 = 0
2x1 + 6x2 + 9x3 = 0
2x1 + 8x2 + 9x3 = 0

after trying to solve this as a linear system of equations problem, i ended getting that each value (x1, x2, x3) was equal to 0... is this anywhere close to a correct approach to this problem? Thanks for any help you have to offer :smile:
 
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Yes, since this matrix is indeed invertible.
 
Do you know about row operations for reducing a matrix? If you do, using them to reduce your matrix would be simpler than solving three equations in three unknowns.
 
My class learned about row operations today but I didn't think to apply them because I began this problem yesterday...

Thanks for your advice :biggrin:
 
If A is invertible, then the only solution to Ax= 0 is x= 0.
 
gpax42 said:
after trying to solve this as a linear system of equations problem, i ended getting that each value (x1, x2, x3) was equal to 0... is this anywhere close to a correct approach to this problem? Thanks for any help you have to offer :smile:

This is a linear system of equation which is equivalent to the matrix representation of the problem. And yes, your solution is correct, as far as I checked.
 

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