Example 2.  Find the inverse of the 5×5 Hilbert matrix  .  

Solution 2.

Enter the matrix using the Table command and the formula for .  

Form the augmented matrix [A, I] and store it in the variable AI.

Construct the inverse matrix using Gauss-Jordan elimination.
Remark.  All the computations are printed too !

Extract the inverse matrix B. from the augmented matrix M = [I,B].

Verify that B is the inverse of A.

Aside.  We can compare our result with Mathematica's built in procedure.  

Aside.  If a numerical answer is required, then we can convert the "exact" matrix to a decimal approximation.

Extract the inverse matrix B. from the augmented matrix M = [I,B].

Verify that B is the inverse of A.

Notice that the final computation contains entries that are "close to zero."  We can chop them off.

Aside.  We can compare our result with Mathematica's built in procedure.  

Remark.  What we do not see is that the numerical calculations were carried out using multiple precision arithmetic and there are about 17 decimal places used in Mathematica's calculations.

Caveat.

    Even for the case of the 5×5 Hilbert matrix we should observe that we are encountering propagation of round off error in the calculations.  

It is easy to determine that the error is getting as large as  .

The next example will illustrate how bad things really are.

(c) John H. Mathews 2004