If $P_1+P_2+\cdots+P_k=I$ and $P_i^2=P_i$ for each $i=1,...,k$ then $P_iP_j=0$ for each $i\neq j$.
I did the following $$P_j=P_jI=P_j\left(\displaystyle\sum_{i=1}^k{}P_i\right)=\displaystyle\sum_{i=1}^k{}P_jP_i=P_j+\displaystyle\sum_{i\neq j}^k{}P_jP_i$$
So, $\displaystyle\sum_{i\neq j}^k{}P_jP_i=0$, Since $P_i$ they are not-negative then $P_jP_i=0$ for each $i\neq j$.
