How do you find the determinant of a 3x3 matrix with polynomial equations?

[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.

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?

 

[Astronuc]

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

Think about

+ - +
- + -
+ - +

 

[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?

 

[kdv]

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.

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?

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.

EDIT: give your answer (before expanded out) and we can check if you made a mistake

 

[Astronuc]

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?

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.