For each positive integer $k$, find the smallest number $n_{k}$ for which there exist real $n_{k} \times n_{k}$ matrices $A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:  
(1) $A_{1}^{2}=A_{2}^{2}=\ldots=A_{k}^{2}=0$,  
(2) $A_{i} A_{j}=A_{j} A_{i}$ for all $1 \leq i, j \leq k$, and  
(3) $A_{1} A_{2} \ldots A_{k} \neq 0$.  
Please provide the sum of the smallest $n_k$ values for $k = 2$ and $k = 3$. 

Let's think step by step and output the final answer within \boxed{}.
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