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

In summary, there is an inconsistency in the application of Cramer's Rule for a 3x3 simple linear matrix, specifically when finding the determinant across the first row. The correct determinant value is 8, but when using the first row, a value of -16 is constantly obtained. The mistake was due to a calculation error.
  • #1
epsilonOri
3
0
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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  • #2
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.
 
  • #3
Omg! Sorry... I calculated incorrectly by forgetting a negative.

My mistake was

1(6-12)

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

Thanks
 
  • #4
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:
[tex]\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[/tex]

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
 
  • #5


I would first check the calculations to ensure that they were done correctly. It is possible that there was a mistake made in the calculations which led to the inconsistency.

If the calculations were correct, then it is possible that there is an error in the application of Cramer's Rule for this specific matrix. Cramer's Rule is based on the assumption that the matrix is invertible, which means that its determinant is non-zero. In this case, the determinant of the matrix across the first row is -16, which is indeed non-zero. However, this does not guarantee that the matrix is invertible.

Upon further investigation, I found that the determinant across the first row is actually the determinant of the transpose of the original matrix. This is because Cramer's Rule is typically applied to the transpose of the matrix, not the original matrix. When the determinant is taken across the transpose of the first row, the correct value of 8 is obtained.

Therefore, it seems that the inconsistency in the application of Cramer's Rule is due to a misunderstanding of the rule itself. It is important to carefully consider the assumptions and conditions for a mathematical rule before applying it to a specific problem. In this case, understanding that Cramer's Rule is applied to the transpose of the matrix would have avoided the confusion and inconsistency.
 

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

1. What is Cramer's Rule?

Cramer's Rule is a mathematical method for solving a system of linear equations by using determinants.

2. How does Cramer's Rule work?

Cramer's Rule works by using determinants to find the unique solution to a system of linear equations. The determinants are calculated using the coefficients of the variables in the equations.

3. What is considered a "weird failure" of Cramer's Rule?

A "weird failure" of Cramer's Rule refers to situations where the determinants in the calculation become zero, resulting in no solution or an infinite number of solutions.

4. What are some possible reasons for the weird failure of Cramer's Rule?

There are several reasons why Cramer's Rule may fail, including when the determinant of the coefficient matrix is zero, when the system of equations is inconsistent or dependent, or when there is a mistake in the calculation.

5. How can I avoid the weird failure of Cramer's Rule?

In order to avoid the weird failure of Cramer's Rule, it is important to check for any mistakes in the calculation and to ensure that the system of equations is consistent. If the determinant of the coefficient matrix is zero, then Cramer's Rule cannot be used and an alternative method for solving the system should be considered.

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