Solving 3 Equations for w in Matrix Form

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Discussion Overview

The discussion revolves around the possibility of expressing three given equations in matrix form, specifically in the style of \( w*u = T*u \), where \( u \) is a vector containing variables \( a, b, c \), and \( T \) is a matrix. The equations involve parameters \( d \) and \( e \) and explore the relationships between these variables.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks if the equations can be expressed in matrix form, specifying the structure \( w*u = T*u \).
  • Another participant inquires about the definitions of \( d \) and \( e \), suggesting that if they are distinct, a different form involving a vector \( v \) may be necessary.
  • A participant proposes an alternative matrix \( T \) that includes terms \( e/c \) and \( d/a \), questioning whether this approach is valid.
  • Another participant reiterates the initial question about the matrix form and reformulates the equations, suggesting a specific matrix representation involving \( 2+w \) and other coefficients.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate form of the matrix \( T \) and the treatment of the variables \( d \) and \( e \). There is no consensus on the best approach to represent the equations in matrix form.

Contextual Notes

Participants have not fully clarified the roles of \( d \) and \( e \) in the equations, and there are unresolved questions regarding the assumptions behind the proposed matrix forms.

andybham
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Is it possible to put these three equations into a matrix equation

w*a = b -2a + e
w*b = c -2b + a
w*c = d -2c + b

of the style w*u = T*u where u is a vector, u = {a, b, c} and T is a matrix. w is just a number.
 
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What are d and e? If they are really different and not typos, then you will need T*u+v=w*u, where v is the vector (e,0,d) and ------------------------------------------------
------------------------------------------------------------------
T=
-2 1 0
1 -2 1
0 1 -2
--------------------------------------------------------------------------------
 
Last edited:
no they are not typos, could I not use

T = -2 1 e/c
1 -2 1
d/a 1 -2

or is that just stupid?
 
andybham said:
Is it possible to put these three equations into a matrix equation

w*a = b -2a + e
w*b = c -2b + a
w*c = d -2c + b

of the style w*u = T*u where u is a vector, u = {a, b, c} and T is a matrix. w is just a number.

What exactly are the unknowns in your equations? a, b, c? If so, then

(2 + w)a - b = e
-a + (2 + w)b - c = 0
-b + (2 + w)c = d ,

which can be written as
[tex]\left(\begin{array}{ccc}2+w & -1 & 0\\-1 & 2+w & -1\\0 & -1 & 2+w\end{array}\right) \left(\begin{array}{ccc}a \\b\\c\end{array}\right)=\left(\begin{array}{ccc}e \\0\\d\end{array}\right)[/tex].
 

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