Why Is a Matrix Not Invertible When Its Determinant Is Zero?

[Jin314159]

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

 

[matt grime]

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.

 

[HallsofIvy]

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.

 

[Jin314159]

Thanks guys for both the geometric and algebraic intuition.

 

[PRodQuanta]

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

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