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Determinant=0 and invertibility |
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| Mar23-04, 05:14 AM | #1 |
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Determinant=0 and invertibility
Can someone provide an intuitive understanding of why a matrix is not invertible when it's determinant is zero?
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| Mar23-04, 05:23 AM | #2 |
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Recognitions:
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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 2-d or 1-d 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. |
| Mar23-04, 09:21 AM | #3 |
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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. |
| Mar23-04, 07:27 PM | #4 |
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Determinant=0 and invertibility
Thanks guys for both the geometric and algebraic intuition.
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| Apr2-04, 11:34 PM | #5 |
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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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