Why Does Cramer's Rule Give Different Determinants for the Same Matrix?

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Discussion Overview

The discussion revolves around the application of Cramer's Rule to a specific 3x3 linear matrix and the resulting determinants calculated from different rows. Participants explore the discrepancies in the determinant values obtained through various methods of calculation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant reports an inconsistency in calculating the determinant for the matrix used in Cramer's Rule, obtaining -16 when expanding across the first row, while getting 8 when using other rows or columns.
  • Another participant requests a step-by-step calculation to understand the discrepancy, noting they consistently get 8 regardless of the minor chosen.
  • A participant acknowledges a calculation error related to a sign, clarifying that their initial determinant calculation was incorrect due to forgetting a negative sign.
  • A later reply questions the interpretation of "taking the determinant across a row," providing a detailed calculation that confirms the determinant is 8, while seeking clarification on the original intent of the question.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the original claim of inconsistency, as one participant corrects their mistake while others maintain differing interpretations of the determinant calculation process.

Contextual Notes

There are unresolved aspects regarding the interpretation of determinant calculations and the specific methods used by participants, which may depend on their understanding of Cramer's Rule and matrix operations.

epsilonOri
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This is very weird, but I found an inconsistency in the application of Cramer's Rule for a 3x3 simple linear matrix.


1x + 1y + 0z = 3
-1x + 3y + 4z = -3
0x + 4y + 3z = 2

Dz =
1 1 3
-1 3 -3
0 4 2

If you take the determinant across the first row To find Dz, I constantly get -16

If you take the determinant across any other rows or columns, you get the correct Dz = 8

What is going on?

Help please.
 
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Not really going to be able to help without seeing a step by step calculation. I get 8 no matter which minor I choose to expand by.
 
Omg! Sorry... I calculated incorrectly by forgetting a negative.

My mistake was

1(6-12)

It should have been
1(6-(-12))

Thanks
 
epsilonOri said:
This is very weird, but I found an inconsistency in the application of Cramer's Rule for a 3x3 simple linear matrix.


1x + 1y + 0z = 3
-1x + 3y + 4z = -3
0x + 4y + 3z = 2

Dz =
1 1 3
-1 3 -3
0 4 2

If you take the determinant across the first row To find Dz, I constantly get -16

If you take the determinant across any other rows or columns, you get the correct Dz = 8

What is going on?

Help please.


What do you mean by "to take the determinant across a row? Do you mean to calculate it wrt the
minors determined by that row? Let's see:
\left|\begin{array}{rrr}1&1&3\\-1&3&-3\\0&4&2\end{array}\right|= 1\cdot\left|\begin{array}{rr}\,3&-3\\\,4&2\end{array}\right|+(-1)\cdot\left|\begin{array}{rr}-1&-3\\0&2\end{array}\right|+3\cdot\left|\begin{array}{rr}-1&3\\0&4\end{array}\right|=(6+12)-(-2)+3(-4)=18+2-12=8

If you meant the above then the result is 8, which is hardly surprising as this is the matrix's determinant ; if you

meant something else then I can't say.

DonAntonio
 

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