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systems of equations with matrices; data fitting 175In many cases, the a and b values are known, so your exercise is to solve forall the x values, taking a as the Hilbert matrix and b as its first row:⎛ 1 1 11 ⎞12 3 4···100[ ]1 1 1 1112 3 4 5··· 101[a ij ]=a ==i + j − 1⎜ . ,..⎟⎝⎠1100[ ] 1[b i ]=b =i11101··· ···199⎛ ⎞1121=3..⎜ ..⎟⎝ ⎠1100Compare to the analytic solution⎛ ⎞ ⎛ ⎞y 1 1y 2⎜ .⎝ .. ⎟⎠ = 0⎜. ⎝ .. ⎟⎠ .y N 08.3.5 Matrix Solution of the String ProblemWe have now set up the solution to our problem of two masses on a string and havethe matrix tools needed to solve it. Your problem is to check out the physical reasonablenessof the solution for a variety of weights and lengths. You should checkthat the deduced tensions are positive and that the deduced angles correspond toa physical geometry (e.g., with a sketch). Since this is a physics-based problem, weknow that the sine and cosine functions must be less than 1 in magnitude and thatthe tensions should be similar in magnitude to the weights of the spheres.8.3.6 Explorations1. See at what point your initial guess gets so bad that the computer is unableto find a physical solution.2. A possible problem with the formalism we have just laid out is that byincorporating the identity sin 2 θ i + cos 2 θ i =1into the equations we may be−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 175