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Determinant=0 and invertibility 
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#1
Mar2304, 05:14 AM

P: n/a

Can someone provide an intuitive understanding of why a matrix is not invertible when it's determinant is zero?



#2
Mar2304, 05:23 AM

Sci Advisor
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P: 9,396

The determinant measures how the volume of the unit box changes. Unit box here means all the points
{(a,b,c...,d)  0<= a,b, ..d <=1 Determinant zero means that it gets squished into smaller dimenisions: eg, for 2x2, the unit square gets sent to a line segment, in 3x3 the unit cube gets sent to either a 2d or 1d figure you can't undo these operations, because infinitely many points get sent to the same place. eg 1 0 0 0 sends all the points with the same y coordinate to the same place, and it squashes the unit square to the unit interval. Is that ok? That's the geometry, we can talk algebraic reasons too. 


#3
Mar2304, 09:21 AM

Math
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Thanks
PF Gold
P: 39,497

A very good "intuitive reason" is that det(AB)= det(A)det(B).
If AB= I then det(A)det(B)= 1 not 0 so neither det(A) nor det(B) can be 0. 


#4
Mar2304, 07:27 PM

P: n/a

Determinant=0 and invertibility
Thanks guys for both the geometric and algebraic intuition.



#5
Apr204, 11:34 PM

P: 354

To find a inverse matrix, you must take 1/det. If your det is equal to zero, it is undifined.
Paden Roder 


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