Question

Some matrix programs, such as MATLAB, have a command to create Hilbert matrices of various sizes. If possible, use an inverse command to compute the inverse of a twelfth-order or larger Hilbert matrix, A. Compute \(A{A^{ - 1}}\). Report what you find.

Short answer

The output matrix is an identity matrix.

Step-by-step solution

Create a Hilbert matrix of large size

Use the MATLAB command to create theHilbert matrix of size\(12 \times 12\).

\( > > {\rm{A}} = {\rm{hilb}}\left( {12} \right)\)

\(\left[ {\begin{array}{*{20}{c}}{1.000}&{0.500}&{0.333}&{0.250}&{0.200}&{0.167}&{0.143}&{0.125}&{0.111}&{0.100}&{0.091}&{0.083}\\{0.500}&{0.333}&{0.250}&{0.200}&{0.167}&{0.143}&{0.125}&{0.111}&{0.100}&{0.091}&{0.083}&{0.077}\\{0.333}&{0.250}&{0.200}&{0.167}&{0.143}&{0.125}&{0.111}&{0.100}&{0.091}&{0.083}&{0.077}&{0.071}\\{0.250}&{0.200}&{0.167}&{0.143}&{0.125}&{0.111}&{0.100}&{0.091}&{0.083}&{0.077}&{0.071}&{0.067}\\{0.200}&{0.167}&{0.143}&{0.125}&{0.111}&{0.100}&{0.091}&{0.083}&{0.077}&{0.071}&{0.067}&{0.063}\\{0.167}&{0.143}&{0.125}&{0.111}&{0.100}&{0.091}&{0.083}&{0.077}&{0.071}&{0.067}&{0.063}&{0.059}\\{0.143}&{0.125}&{0.111}&{0.100}&{0.091}&{0.083}&{0.077}&{0.071}&{0.067}&{0.063}&{0.059}&{0.056}\\{0.125}&{0.111}&{0.100}&{0.091}&{0.083}&{0.077}&{0.071}&{0.067}&{0.063}&{0.059}&{0.056}&{0.053}\\{0.111}&{0.100}&{0.091}&{0.083}&{0.077}&{0.071}&{0.067}&{0.063}&{0.059}&{0.056}&{0.053}&{0.050}\\{0.100}&{0.091}&{0.083}&{0.077}&{0.071}&{0.067}&{0.063}&{0.059}&{0.056}&{0.053}&{0.050}&{0.048}\\{0.091}&{0.083}&{0.077}&{0.071}&{0.067}&{0.063}&{0.059}&{0.056}&{0.053}&{0.050}&{0.048}&{0.045}\\{0.083}&{0.077}&{0.071}&{0.067}&{0.063}&{0.059}&{0.056}&{0.053}&{0.050}&{0.048}&{0.045}&{0.043}\end{array}} \right]\)

Obtain the inverse of matrix A

Compute theinverse of matrix A by using the MATLAB command shown below:

\( > > B = {\rm{A}}\^ - 1\)

\[\left[ {\begin{array}{*{20}{c}}{1.41e + 02}&{ - 9.93e + 05}&{2.29e + 05}&{ - 2.54e + 06}&{1.61e + 07}&{ - 6.35e + 07}&{1.62e + 08}\\{ - 9.93e + 03}&{9.35e + 05}&{ - 2.43e + 07}&{2.89e + 08}&{ - 1.91e + 09}&{7.76e + 09}&{ - 2.03e + 10}\\{2.29e + 05}&{ - 2.43e + 07}&{6.75e + 08}&{ - 8.38e + 09}&{5.72e + 10}&{ - 2.37e + 11}&{6.29e + 11}\\{ - 2.54e + 06}&{2.89e + 08}&{ - 8.38e + 09}&{1.07e + 11}&{ - 7.48e + 11}&{3.15e + 12}&{ - 8.48e + 12}\\{1.61e + 07}&{ - 1.91e + 09}&{5.72e + 10}&{ - 7.48e + 11}&{5.30e + 12}&{ - 2.27e + 13}&{6.16e + 13}\\{ - 6.35e + 07}&{7.76e + 09}&{ - 2.37e + 11}&{3.15e + 12}&{ - 2.27e + 13}&{6.16e + 13}&{ - 2.69e + 14}\\{1.62e + 08}&{ - 2.03e + 10}&{6.29e + 11}&{ - 8.48e + 12}&{6.16e + 13}&{ - 2.69e + 14}&{7.44e + 14}\\{ - 2.73e + 08}&{3.47e + 10}&{ - 1.09e + 12}&{1.49e + 13}&{ - 1.09e + 14}&{4.80e + 14}&{ - 1.34e + 15}\\{3.02e + 08}&{ - 3.89e + 10}&{1.24e + 12}&{ - 1.70e + 13}&{1.26e + 14}&{ - 5.57e + 14}&{1.56e + 15}\\{ - 2.10e + 08}&{2.74e + 10}&{ - 8.81e + 11}&{1.22e + 13}&{ - 9.08e + 13}&{4.04e + 14}&{ - 1.14e + 15}\\{8.83e + 07}&{ - 1.10e + 10}&{3.57e + 11}&{ - 4.98e + 12}&{3.72e + 13}&{ - 1.66e + 14}&{4.70e + 14}\\{ - 1.45e + 07}&{1.93e + 09}&{ - 6.29e + 10}&{8.82e + 11}&{ - 6.63e + 12}&{2.98e + 13}&{ - 8.44e + 13}\end{array}} \right.\]

\(\left. {\begin{array}{*{20}{c}}{ - 2.73e + 08}&{3.02e + 08}&{ - 2.10e + 08}&{8.38e + 07}&{ - 1.45e + 07}\\{3.47e + 10}&{ - 3.89e + 10}&{ - 2.74e + 10}&{ - 1.10e + 10}&{1.93e + 09}\\{ - 1.09e + 12}&{1.24e + 12}&{ - 8.81e + 11}&{3.57e + 11}&{ - 6.29e + 10}\\{1.49e + 13}&{ - 1.70e + 13}&{1.22e + 13}&{ - 4.89e + 12}&{8.82e + 11}\\{ - 1.09e + 14}&{1.26e + 14}&{ - 9.08e + 13}&{3.72e + 13}&{ - 6.63e + 12}\\{4.80e + 14}&{ - 5.57e + 14}&{4.04e + 14}&{ - 1.66e + 14}&{2.98e + 13}\\{ - 1.34e + 15}&{1.56e + 15}&{ - 1.14e + 15}&{4.70e + 14}&{ - 8.44e + 13}\\{2.42e + 15}&{ - 2.84e + 15}&{2.80e + 15}&{ - 8.62e + 14}&{1.55e + 14}\\{ - 2.84e + 15}&{3.34e + 15}&{ - 2.45e + 15}&{1.02e + 15}&{ - 1.85e + 14}\\{2.08e + 15}&{ - 2.45e + 15}&{1.81e + 15}&{ - 7.56e + 14}&{1.37e + 14}\\{ - 8.62e + 14}&{1.02e + 15}&{ - 7.56e + 14}&{3.17e + 14}&{ - 5.75e + 13}\\{1.55e + 14}&{ - 1.85e + 14}&{1.37e + 14}&{ - 5.75e + 13}&{1.05e + 13}\end{array}} \right]\)

Obtain the product of the matrix and its inverse

Compute matrix C by using the MATLAB command shown below:

\( > > C = {\rm{A}}*{\rm{B}}\)

\(\left[ {\begin{array}{*{20}{c}}{1.000}&{0.000}&{0.000}&{ - 0.001}&{0.001}&{0.006}&{ - 0.010}&{ - 0.019}&{ - 0.054}&{0.000}&{0.007}&{ - 0.001}\\{ - 0.005}&{1.000}&{0.000}&{0.000}&{0.002}&{0.000}&{0.020}&{ - 0.052}&{0.025}&{ - 0.019}&{0.003}&{ - 0.004}\\{ - 0.006}&{0.003}&{1.000}&{0.000}&{0.002}&{ - 0.002}&{0.023}&{ - 0.042}&{0.041}&{ - 0.036}&{0.007}&{ - 0.003}\\{ - 0.007}&{0.004}&{ - 0.001}&{1.000}&{0.002}&{ - 0.003}&{0.015}&{ - 0.035}&{0.049}&{ - 0.047}&{0.003}&{ - 0.003}\\{ - 0.007}&{0.006}&{ - 0.001}&{0.000}&{1.003}&{0.000}&{0.001}&{ - 0.029}&{0.027}&{ - 0.021}&{0.006}&{ - 0.002}\\{ - 0.007}&{0.006}&{ - 0.001}&{ - 0.001}&{0.002}&{0.997}&{0.020}&{ - 0.036}&{0.047}&{ - 0.036}&{0.010}&{ - 0.003}\\{ - 0.007}&{0.006}&{ - 0.001}&{ - 0.002}&{0.002}&{ - 0.004}&{1.012}&{ - 0.040}&{0.047}&{ - 0.034}&{0.010}&{ - 0.002}\\{ - 0.006}&{0.007}&{0.000}&{ - 0.003}&{0.002}&{ - 0.006}&{0.016}&{0.948}&{0.050}&{ - 0.040}&{0.015}&{ - 0.003}\\{ - 0.006}&{0.007}&{0.000}&{ - 0.003}&{0.001}&{0.005}&{ - 0.011}&{0.012}&{1.012}&{ - 0.001}&{ - 0.001}&{ - 0.001}\\{ - 0.006}&{0.006}&{0.000}&{ - 0.004}&{0.003}&{ - 0.002}&{0.005}&{ - 0.031}&{0.049}&{0.975}&{0.13}&{ - 0.002}\\{ - 0.006}&{0.006}&{0.001}&{ - 0.004}&{0.000}&{0.009}&{ - 0.022}&{0.009}&{ - 0.033}&{0.025}&{0.989}&{0.002}\\{ - 0.006}&{0.006}&{0.001}&{ - 0.005}&{0.002}&{0.001}&{0.006}&{ - 0.018}&{0.025}&{ - 0.018}&{0.002}&{0.999}\end{array}} \right]\)

Thus, the resultant matrix is an identity matrix.