Determinant of a 3X3 matrix

1. Feb 16, 2008

snoggerT

M= |3-2x^3_____-4+2x^2+4x^3______0|
|-x^3________1+x^2+2x^3_______0|
|-8-6x^2_____16+12x^2____-1-3x^2|

find the determinant.

* I put the underscores for spacing.

3. The attempt at a solution

I first tried finding det(A) as if it was a regular matrix of numbers, but that doesn't seem to work. So I'm really not sure how to find the determinant for a matrix of polynomial equations. Where would you actually start?

2. Feb 16, 2008

Staff: Mentor

Well, one can use the fact that the third column has two zeros as entries.

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3. Feb 16, 2008

snoggerT

I'm not quite getting what I'm supposed to see from the 2 zeros. The way I worked it out before was using the normal determinant method of a11(a22*a33-a32*a23)-a12...etc. Would that not be the correct way to find a determinant for this type of matrix?

4. Feb 16, 2008

kdv

It should work! If it does not work, check your calculation. It's really the same as the determinant of a matrix of numbers. The fact that there are two zeroes will make the final result not too awful looking. But it's the usual formula for the determinant of a 3 by 3 matrix.

Last edited: Feb 16, 2008
5. Feb 16, 2008

Staff: Mentor

That would be one way. The co-factor expansion (or determinant expansion by minors) can be done on any column or row! If one does is using the third column, one sees a33|A33|, where A33 is the 3,3 minor.

http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html

See the examples here - http://en.wikipedia.org/wiki/Laplace_expansion - using first row or second column. One can also use the third row or third column.