Calculating Matrix Determinant of 4x4: Different Answers

[sibiryk]

I have a matrix 4x4:

1,2,3,4
2,6,6,6
0,0,1,13
0,0,0,1

I need to find determinant. I am getting det=6.
When I use online matrix calculator it gives me det=2.

I tried reduce second row (2 row - 1(1 row)) and I got det=2

I'm confused. Why calculator gives me different answer?

 

[Hurkyl]

So, by what method did you get 6, and why do you think the method is appropriate?

 

[sibiryk]

I used formula (10) from
here http://mathworld.wolfram.com/Determinant.html

 

[Hurkyl]

You forgot to answer the second half of my question. Why do you think that formula is appropriate for your problem?

(Or, to put it differently, when is that formula applicable, and is your problem one of those times?)

 

[sibiryk]

Actually I'm not sure about that. Do you want to say that this formula is not aplicatable for 4x4 matrix?

 

[Hurkyl]

I didn't want to say it -- I was hoping that it would be obvious from the fact that the matrix in (10) is a 3x3 matrix, and that you would see it for yourself.

I suspect, despite the fact you have not said this is what you've done, that you attempted to make up your own generalization of that formula to 4x4 matrices, and got it wrong.

(Incidentally, the text preceeding formula (10), of which (10) is an example, does describe a correct way to compute a determinant, but it's not a practical method)

 

[amcavoy]

My suggestion is (if you are going to use the cofactor method) that you take the determinant across the bottom row. It has the most zeroes and you will find it easiest to compute.

\det{A}=\sum_{j=1}^{4}\left(-1\right)^{j+4}\det{A_{4j}}

...where det(A_4j) is the determinant of the matrix after deleting row 4 and column j.

 

[TeTeC]

From this point, the easiest method to find this determinant would be with a gaussian elimination. It would take only one operation...

 

[HallsofIvy]

Indeed, that is what made sibiryk's post so puzzling! According to him he did it one way and got an answer of 6, then row-reduce the matrix to a triangular form- which gave an answer of 2, the same as he got from the calculator and was wondering why the calculator said 2!

 

[sibiryk]

Yes, I was talking about gaussian elimination and I made a mistake.
It was suppouse to be (2nd row -2(1st row)).

Thank you guys very much!