Module for Matrix Inverse

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Theorem (Inverse Matrix) Assume that is an nonsingular matrix. Form the augmented matrix of dimension  .  Use Gauss-Jordan elimination to reduce the matrix so that the identity is in the first columns.  Then the inverse is located in columns .  The augmented matrix looks like:

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    by using Gauss-Jordan elimination.  Provision is made for row interchanges if they are needed.

Mathematica Subroutine (Complete Gauss-Jordan Elimination).

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Example 1. Use Gauss-Jordan elimination to find the inverse of the matrix  .
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.

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.

(c) John H. Mathews 2003