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

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A matrix is not invertible when its determinant is zero because the determinant indicates how the volume of a geometric shape changes under transformation. A zero determinant implies that the transformation squashes the shape into a lower dimension, making it impossible to uniquely map back to the original points. For example, a 2x2 matrix with a determinant of zero collapses a unit square into a line segment, losing information about the original points. Algebraically, if the product of two matrices equals the identity matrix, their determinants must also multiply to one, indicating neither can be zero. Therefore, a zero determinant leads to an undefined inverse, confirming the matrix's non-invertibility.
Jin314159
Can someone provide an intuitive understanding of why a matrix is not invertible when it's determinant is zero?
 
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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.
 
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.
 
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Likes MathewsMD
Thanks guys for both the geometric and algebraic intuition.
 
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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