I think I found a typo in the final line (not impactful for the result, however).

11/89 should be 11/8900

This can be verified by using the following Sage code (simply go to https://sagecell.sagemath.org/ to avoid installing sage)

  b = 10

  A = Matrix([[0, 1],[1/(b**2), 1/b]])
  I = matrix.identity(2)

  show((I - A).inverse() * Matrix([[1/b**2],[1/b**3]]))