1. The Hilbert matrices are defined by These matrices are famous because they are ill-conditioned. A small change in a matrix entry or the right-hand side vector can cause the system to have a very...

1. The

Hilbert matrices are defined by

These matrices are famous because they are ill-conditioned. A small change in a matrix entry or the right-hand side vector can cause the system

to have a very different solution. We will study ill-conditioning in Chapter 10. For now, we will look at the determinants of Hilbert matrices and their inverses. The MATLAB command hilb(n) builds the

Hilbert matrix.

Compute the determinant of the Hilbert matrices of order 5, 10, 15, and 25. What appears to happen as the order increases?

It can be shown that the inverse of a Hilbert matrix consists entirely of integers. Compute the inverse of the Hilbert matrix of order

Compute the determinant of the inverse for each of the Hilbert matrices of order

and

Relate your results to those of part