# Weird failure of Cramers Rule

1. Aug 10, 2012

### 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?????????

2. Aug 10, 2012

### 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.

3. Aug 10, 2012

### epsilonOri

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

My mistake was

1(6-12)

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

Thanks

4. Aug 10, 2012

### DonAntonio

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