The n × n matrix Hn whose i jth entry is 1/(i + j − 1) is called the Hilbert matrix of order n.
(a) Write out H2, H3, H4, H5, and H6. Use a computer to calculate their determinants exactly. What seems to happen to det Hn as n gets larger?
(b) Now calculate H10 and det H10. If you use exact arithmetic, you should find that det H10 0 and hence that H10 is invertible. (See Exercises 30–38 of §1.6 for more about invertible matrices.)
(c) Now give a numerical approximation A for H10. Calculate the inverse matrix B of this approximation, if your computer allows. Then calculate AB and BA. Do you obtain the 10 × 10 identity matrix I10 in both cases?
(d) Explain what parts (b) and (c) suggest about the difficulties in using numerical approximations in matrix arithmetic.
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