In 5a, the eigenvectors are obtained pretty quickly by guessing. I wouldn't expect you to converge on the eigenvectors this quickly. Probably several pages of multiplications and guessing to build your intuition is more realistic. Questions 5 and 6 are intended to build your familiarity with matrix multiplication, and eigenvector and e.v. identification, since that will be important in the latter part of this course.
In problem 9, some interesting issues arise. There is a crossover from a region of low transmission at low energy, to a region of high transmission for and incident electron of high-energy. The relationship of that to length scale (wavelength) and energy, and the phase-shift behavior are well worth understanding if you plan to take 139b and/or go to physics grad school. They care a lot about phase shifts and understanding crossover scales there. I only calculated C/A, which represents the transmission coefficient. You can do B/A, the reflectivity coefficient, by the same method and it complements C/A in an intuitive way. (What relation links them?)
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