When is Matrix C Not Invertible?

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Discussion Overview

The discussion revolves around the conditions under which a given matrix C is not invertible, specifically focusing on the calculation of its determinant and the implications of scaling the matrix. Participants explore the determinant's value, the relationship between the determinant of a scaled matrix, and the specific values of 'x' that lead to non-invertibility.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant calculates the determinant of matrix C as x^2 - 12x + 27 and suggests that C is not invertible when this determinant equals zero, proposing x = 9 or 3 as solutions.
  • Another participant agrees with the determinant calculation and suggests that the determinant of a scaled matrix follows the rule det(aC) = a^N det(C), proposing that for a 3x3 matrix, det(2C) = 8 det(C).
  • A later reply confirms the previous answers and expresses appreciation for the clarification.
  • Another participant provides an example using the identity matrix to illustrate the determinant of a scaled matrix, reinforcing the earlier claim about the determinant's behavior under scaling.

Areas of Agreement / Disagreement

Participants generally agree on the calculations of the determinant and the conditions for non-invertibility. However, there is some uncertainty regarding the terminology used to describe the dimension of the matrix and the scaling of the determinant.

Contextual Notes

Some participants express uncertainty about the terminology related to matrix dimensions and scaling, which may affect the clarity of their arguments.

dcl
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<br /> c = \left[ {\begin{array}{*{20}c}<br /> {2 - x} &amp; 5 &amp; 1 \\<br /> { - 3} &amp; 0 &amp; x \\<br /> { - 2} &amp; 1 &amp; 2 \\<br /> \end{array}} \right]

a) Calculate det(C).
My answer was x^2 - 12x + 27.

b) Calculate det(2C).
Umm, would this just be 2*det(C)?
Couldn't find anything more helpful in my notes.

c) State the values for 'x' for which C is not invertible.
I believe the value for 'x' that would make this non invertable would be the solution that det(c) = 0. (A matrix has no inverse when the determinant = 0 yeh?)
which would be x = 9 or 3
is this correct?
 
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I get the same answer as you on (a) and (c). I'm thinking that det(aC)=a^N det(c) where a is a constant and N is the "dimension" (wrong terminology? NxN matrix). If I am right about that, then for a 3x3 matrix, det(2C) = 8 det(C).

You may be thinking about the trace of a matrix: Tr(2C) = 2 Tr(C).
 
Last edited:
Hmm, yeh, what you're saying would make more sense.
Thanks for confirming the other answers.
 
Janitor is correct about (b). To see it, just consider what happens if C is the identity: 2I has 2 along the diagonal and 0 everywhere else, so the determinant is det(2I) = 8 = 23det(I).
 

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