- 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 .