# Determinant=0 and invertibility

1. Mar 23, 2004

### Jin314159

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

2. Mar 23, 2004

### 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.

3. Mar 23, 2004

### HallsofIvy

Staff Emeritus
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.

4. Mar 23, 2004

### Jin314159

Thanks guys for both the geometric and algebraic intuition.

5. Apr 2, 2004

### PRodQuanta

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