Properties of the trace

1. Immediate from the definition: $\operatorname{tr}(A+B) = \sum_i (a_{ii}+b_{ii}) = \sum_i a_{ii} + \sum_i b_{ii}$.

2.
$$\operatorname{tr}(AB) = \sum_i \sum_j a_{ij}b_{ji} = \sum_j \sum_i b_{ji}a_{ij} = \operatorname{tr}(BA).$$

3. Using (2) with $A = P^{-1}A$ and $B = P$:
$$\operatorname{tr}(P^{-1}AP) = \operatorname{tr}((P^{-1}A)P) = \operatorname{tr}(P(P^{-1}A)) = \operatorname{tr}(A).$$

Conclusion. The trace is linear, invariant under cyclic permutations, and invariant under similarity.