\displaystile tr BA=\sum_{k=1}^n d_{kk}=\sum_{k=1}^n(\sum_{i=1}^nb_{ki}a_{ik})=\sum_{k=1}^n(\sum_{i=1}^na_{ik}b_{ki})=\sum_{i=1}^n(\sum_{k=1}^na_{ik}b_{ki})