Determinant of 3x3 Matrix without direct evaluation

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Homework Help Overview

The discussion revolves around finding the determinant of a 3x3 matrix without direct evaluation. The matrix in question includes polynomial expressions, and the goal is to show that its determinant equals \((x^2 + 2)^3\).

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are considering various methods to compute the determinant, including the cofactor method. Questions about the nature and application of cofactors are raised, as well as comparisons to other methods like transforming the matrix into an upper triangular form.

Discussion Status

The discussion is ongoing, with participants exploring different methods and clarifying concepts related to the cofactor approach. There is no explicit consensus yet, but several lines of reasoning are being examined.

Contextual Notes

Participants are constrained by the requirement to avoid direct evaluation of the determinant, which influences their exploration of alternative methods.

ZedCar
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Homework Statement


Show that:

(x^2) (2x) (-2)
(2x) (2-x^2) (2x)
(2) (-2x) (-x^2)

= (x^2 + 2)^3

Do not use direct evaluation.

Homework Equations


The Attempt at a Solution



As direct evaluation is not permitted, I'm wondering which method should I use? Thank you
 
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what possible methods do you have in mind?
 
Is it a method using cofactors?
 
okay, now what does the method of cofactors do?
 
I'm going to have to look into that.

I have a sheet which I just got today from lecture and it states;

Cofactor is the minor multiplied by the sign of the element
Cij = (-1)^(i+j) Mij
 
Is this cofactor method different from making an upper triangular matrix from it?
 

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