Let Hn be the n × n Hilbert matrix (1.70), and Kn = H-ln its inverse. It can be proved, [32, p. 513], that the (i, j) entry of Kn is

Where

is the standard binomial coefficient.
(a) Write down the inverse of the Hilbert matrices H3, H4, H5 by either using the formula or using the Gauss-Jordan Method with exact rational arithmetic. Check your results by multiplying the matrix by its inverse.
(b) Recompute the inverses on your computer using floating point arithmetic and compare with the exact answers.
(c) Try using floating point arithmetic to find k10 and K20. Test the answer by multiplying the Hilbert matrix by its computed inverse.

Transcribed Image Text:

(-1)i+) (it j-1).