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

[epsilonOri]

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.

 

[Muphrid]

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.

 

[epsilonOri]

Omg! Sorry... I calculated incorrectly by forgetting a negative.

My mistake was

1(6-12)

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

Thanks

 

[DonAntonio]

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

 

[blue_raver22]

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.