Simplifying Determinants to Finding Linear Factors - Homework Help

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Homework Help Overview

The problem involves expressing a determinant of a 3x3 matrix as a product of four linear factors. The matrix consists of constants and variables, specifically involving powers of the variables.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss various methods for simplifying the determinant, including row operations and expansion techniques. Some express difficulty in visualizing the factorization and mention extensive attempts without success.

Discussion Status

The discussion is ongoing, with participants sharing different strategies and identities that may assist in the factorization process. There is recognition of the challenges involved, and some guidance has been provided regarding specific algebraic identities.

Contextual Notes

Participants note the complexity of the determinant and the difficulty in presenting their work clearly, particularly when using LaTeX. There is an emphasis on the need for careful manipulation of the matrix to achieve the desired factorization.

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Homework Statement



Express the determinant as a product of four linear factors.

[itex] \left(<br /> \begin{array}{ccc}<br /> 1 & a & a^3 \\<br /> 1 & b & b^3 \\<br /> 1 & c & c^3<br /> \end{array}<br /> \right)[/itex]

I'm sure that the only way to do this without hurting yourself is to operate on the determinant and take factors out. It just doesn't move. I can't see the pattern, I can't see how to get factors out of it. I add rows, columns everything.
 
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Expand along the first row and then just try to factorize it.
 
rock.freak667 said:
Expand along the first row and then just try to factorize it.

Can't do it. It's hard to show my working on latex buts there's pages of attempts. x^3 I just can't do.
 
Try row reducing once, with R2-R1 and R3-R1. Then expand along row one.


then use this identity and see if it helps a3-b3=(a-b)(a2+ab+b2)
 
rock.freak667 said:
Try row reducing once, with R2-R1 and R3-R1. Then expand along row one.


then use this identity and see if it helps a3-b3=(a-b)(a2+ab+b2)


Excellent thanks, that's the identity I needed.
 

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