Matrix, Find determinant using properties of Det.

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Homework Help Overview

The problem involves finding the determinant of a 3x3 matrix using properties of determinants. The matrix is presented alongside an equation that represents its determinant in terms of its elements.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the properties of determinants but finds no similar problems in their readings. They express a need for assistance. One participant suggests a method involving column operations and expansion by minors.

Discussion Status

Some participants have reported solving the problem independently after engaging with the suggestions provided. However, there is no explicit consensus on the methods used or the reasoning behind them.

Contextual Notes

The original poster indicates a lack of similar problems in their study materials, which may affect their approach to the problem. There are also references to specific transformations of the matrix that lead to the determinant's factorization.

am_knightmare
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Homework Statement


1 1 1
a b c = (b-a)(c-a)(c-b)
a^2 b^2 c^2
(above is a 3x3 matrix equaling to a equation)
question:"Show by applying property of the determinant"

Homework Equations


N/A

The Attempt at a Solution


read through the whole chapter of determinants, there were no similar problems. read it the second time focusing on properties, no simimlar properties, Please help.
 
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Start by subtracting the first column from the second and third columns. Then think about an expansion by minors.
 
Just solved it. thanks for the reply though.
1 1 1
a b c
a^2 b^2 c^2
becomes
1 0 0
a b-a c-a
a^2 (b^2-a^2) (c^2-a^2)
then becomes
(b-a)(c-a) times
1 0 0
a 1 1
a^2 b-a c-a
then
1 0 0
a 1 0
a^2 b-a c-b
det= 1 x 1 x(c-b) ( c-a) (b-a)
 
am_knightmare said:
Just solved it. thanks for the reply though.
1 1 1
a b c
a^2 b^2 c^2
becomes
1 0 0
a b-a c-a
a^2 (b^2-a^2) (c^2-a^2)
then becomes
(b-a)(c-a) times
1 0 0
a 1 1
a^2 b-a c-a
then
1 0 0
a 1 0
a^2 b-a c-b
det= 1 x 1 x(c-b) ( c-a) (b-a)

Yup. That'll do it.
 

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