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

I think the second column is the "nicest" for your problem since you see the two zeros in the third column - making the first term in the expansion simple.In summary, the determinant of the given matrix can be found by using the co-factor expansion method and choosing the second column. The two zeros in the third column make the first term in the expansion simpler. The result will be a polynomial expression.
  • #1
snoggerT
186
0
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?
 
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  • #2
Well, one can use the fact that the third column has two zeros as entries.

Think about

+ - +
- + -
+ - +
 
  • #3
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
snoggerT said:
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
 
Last edited:
  • #5
snoggerT said:
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.
 

What is a determinant of a 3X3 matrix?

A determinant of a 3X3 matrix is a numerical value that can be calculated from the elements of the matrix. It provides important information about the matrix, such as whether it is invertible and the scaling factor of the transformation described by the matrix.

How do you calculate the determinant of a 3X3 matrix?

The determinant of a 3X3 matrix can be calculated using various methods such as cofactor expansion, row reduction, or using the Sarrus rule. The most common method is using cofactor expansion, where the matrix is broken down into smaller matrices and their determinants are calculated.

Why is the determinant of a 3X3 matrix important?

The determinant of a 3X3 matrix is important because it provides crucial information about the matrix, such as its invertibility and the scaling factor of the transformation described by the matrix. It is also used in various mathematical applications, such as solving systems of linear equations and calculating areas and volumes.

What is the relationship between the determinant and the eigenvalues of a 3X3 matrix?

The determinant of a 3X3 matrix is equal to the product of its eigenvalues. This means that by calculating the determinant, we can determine the eigenvalues of the matrix. Additionally, the determinant can also help determine whether a matrix has distinct eigenvalues or not.

How does changing the elements of a 3X3 matrix affect its determinant?

Changing the elements of a 3X3 matrix can have a significant impact on its determinant. For example, swapping two rows or columns will result in a determinant with the opposite sign. Additionally, multiplying a row or column by a scalar will result in the determinant being multiplied by the same scalar. These changes can ultimately affect the properties and behavior of the matrix.

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