The n × n matrix Hn whose i jth entry is 1/(i + j − 1) is called the Hilbert matrix of o...

The n × n matrix H_n 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 H_n 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.