Square matrix/Characteristic polynomial splits/Trace/Exercise

Let ${}K$ be a field, and let ${}M$ denote an ${}n\times n$-matrix over ${}K$ with the property, that the characteristic polynomial splits into linear factors, that is,

${}\chi _{M}=(X-\lambda _{1})^{\mu _{1}}\cdot (X-\lambda _{2})^{\mu _{2}}\cdots (X-\lambda _{k})^{\mu _{k}}\,.$

Show that

${}\operatorname {trace} {\left(M\right)}=\sum _{i=1}^{k}\mu _{i}\lambda _{i}\,.$

Create a solution