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Matrix with No Inverse

Date: 06/13/2005 at 09:55:27
From: Saul
Subject: Solving 3x3 matrix

Are there any 3x3 matrices that do not have inverses and if so, where 
do I find them?



Date: 06/13/2005 at 14:31:17
From: Doctor Samuel
Subject: Re: Solving 3x3 matrix

Hi Saul,

A matrix has an inverse if and only if its determinant is nonzero.
Thus, for a 3x3 matrix A,

      [ a1 a2 a3 ]
  A = [ b1 b2 b3 ]
      [ c1 c2 c3 ]

The inverse of A doesn't exist iff

  a1*b2*c3 + a2*b3*c1 + a3*b1*c2 - a1*b3*c2 - a3*b2*c1 - a2*b1*c3 = 0. 

From this we can see that A cannot have an inverse if, for example, 

  a1 = a2 = a3 = 0.

Matrices that don't have inverses are called "singular".  Perhaps you
can find other values for which A is singular?

Hope this helps.  Please write back if you are still stuck.

- Doctor Samuel, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Linear Algebra

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