Module for Matrix Inverse

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Theorem (Inverse Matrix) Assume that [Graphics:Images/MatrixInverseMod_gr_1.gif] is an [Graphics:Images/MatrixInverseMod_gr_2.gif] nonsingular matrix. Form the augmented matrix [Graphics:Images/MatrixInverseMod_gr_3.gif] of dimension  [Graphics:Images/MatrixInverseMod_gr_4.gif].  Use Gauss-Jordan elimination to reduce the matrix [Graphics:Images/MatrixInverseMod_gr_5.gif] so that the identity [Graphics:Images/MatrixInverseMod_gr_6.gif] is in the first [Graphics:Images/MatrixInverseMod_gr_7.gif] columns.  Then the inverse [Graphics:Images/MatrixInverseMod_gr_8.gif] is located in columns [Graphics:Images/MatrixInverseMod_gr_9.gif].  The augmented matrix [Graphics:Images/MatrixInverseMod_gr_10.gif] looks like:

    [Graphics:Images/MatrixInverseMod_gr_11.gif]

We can use the previously developed Gauss-Jordan subroutine to find the inverse of a matrix.

Algorithm II. (Complete Gauss-Jordan Elimination).  Construct the solution to the linear system  [Graphics:Images/MatrixInverseMod_gr_12.gif]  by using Gauss-Jordan elimination.  Provision is made for row interchanges if they are needed.  

Mathematica Subroutine (Complete Gauss-Jordan Elimination).

[Graphics:Images/MatrixInverseMod_gr_13.gif]

Example 1. Use Gauss-Jordan elimination to find the inverse of the matrix  [Graphics:Images/MatrixInverseMod_gr_14.gif].  
Solution 1.

 

Example 2. Find the inverse of the 5x5 Hilbert matrix.
Solution 2.

 

Example 3. Hilbert matrices are known to be ill-conditioned. Find the inverse of the 5x5 matrix that approximates the 5x5 Hilbert matrix.
Remark. The entries in the matrix for this exercise must be typed in by hand in order to make sure that only six decimal places are stored in the computer.
Solution 3.

 

 

Old Lab Project (Matrix Inversion  Matrix Inversion).   Internet hyperlinks to an old lab project.  

 

Research Experience for Undergraduates

The Inverse Matrix  The Inverse Matrix  
Internet hyperlinks to web sites and a bibliography of articles.  

The Hilbert Matrix  The Hilbert Matrix  
Internet hyperlinks to web sites and a bibliography of articles.  

 

Downloads (Matrix Inversion Matrix Inversion).  Download this Mathematica notebook.  

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2003