Errors and addenda for MP274 notes

• 19th January 2005. In the section Exponential of a matrix, page 10, the (1,3) entry should be -3 instead of -2.
• 18th January 2019. In section Applications of the Smith normal form, on page 140, Case 2, replace the line before "≥ 0" by
  = {m_k + n_k - 2min(m_k, n_l)} + {m_k + n_k - 2min(m_l, n_k)}

  = 2{m_k + n_k - (min(m_k, n_l) + min(m_l, n_k))} ≥ 0,

  as min(m_k, n_l) ≤ m_k and min(m_l, n_k) ≤ n_k.

• 14th July 2019. In section Markov matrices, in the proof of Theorem 4.10 on page 90, add the explanation .