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

  • Context: Undergrad 
  • Thread starter Thread starter Jin314159
  • Start date Start date
Click For Summary

Discussion Overview

The discussion centers around the concept of matrix invertibility and its relationship with the determinant, particularly focusing on why a matrix is not invertible when its determinant is zero. The scope includes both intuitive geometric interpretations and algebraic reasoning.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that the determinant represents how the volume of a unit box changes, with a determinant of zero indicating that the box is squished into lower dimensions, making it impossible to invert the transformation.
  • One participant illustrates this idea with examples, such as a 2x2 matrix transforming a unit square into a line segment, suggesting that multiple points collapse to the same location.
  • Another participant mentions the property that if the product of two matrices equals the identity matrix (AB = I), then the determinants must satisfy det(AB) = det(A)det(B) = 1, implying that neither determinant can be zero.
  • A later reply emphasizes that finding an inverse matrix involves taking the reciprocal of the determinant, which is undefined when the determinant is zero.

Areas of Agreement / Disagreement

Participants generally agree on the relationship between the determinant and matrix invertibility, though they present different perspectives and interpretations. There is no explicit consensus on a singular intuitive explanation.

Contextual Notes

The discussion includes both geometric and algebraic reasoning, but does not resolve the nuances of these interpretations or their implications for understanding matrix properties.

Who May Find This Useful

This discussion may be useful for students and practitioners in mathematics and engineering who are exploring the concepts of linear algebra, particularly those interested in matrix theory and determinants.

Jin314159
Can someone provide an intuitive understanding of why a matrix is not invertible when it's determinant is zero?
 
Physics news on Phys.org
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.
 
  • Like
Likes   Reactions: 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
 

Similar threads

Graduate What causes a complex symmetric matrix to change from invertible to non-invertible?
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Matrix rank in terms of determinant polynomial
  • · Replies 14 ·
Replies
14
Views
5K
Undergrad Invertible Matrix Proof: A-Transpose * M * A (n by m)
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Spectral theorem for Hermitian matrices-- special cases
  • · Replies 2 ·
Replies
2
Views
4K
MHB 5.3 Show that a square matrix with a zero row is not invertible.
  • · Replies 3 ·
Replies
3
Views
2K
MHB Proving or disproving this matrix V is invertible.
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Similarity transformation changing the determinant to 1
  • · Replies 20 ·
Replies
20
Views
4K
Undergrad Why Use Gram-Schmidt to Make a Unitary Matrix?
  • · Replies 14 ·
Replies
14
Views
5K
MHB Proving A is Zero Matrix if B is Invertible & Same Size as A
  • · Replies 1 ·
Replies
1
Views
2K
MHB Determinant always equal to zero?
  • · Replies 5 ·
Replies
5
Views
3K