Determinant=0 and invertibility

  1. Can someone provide an intuitive understanding of why a matrix is not invertible when it's determinant is zero?
  2. jcsd
  3. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    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.


    |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.
  4. HallsofIvy

    HallsofIvy 41,260
    Staff Emeritus
    Science Advisor

    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.
    MathewsMD likes this.
  5. Thanks guys for both the geometric and algebraic intuition.
  6. 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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