 Solution

Use row vectors for population counts. Given $Z_t$, independent reproduction has conditional mean $\mathbb{E}[Z_{t+1}\mid Z_t]=Z_t\Lambda$. Starting from $Z_0=e_i^T$, induction gives $\mathbb{E}{Z}_t=e_i^T{\Lambda }^t$. Summing all child types, the expected size of generation $t$ isThis also covers $t=0$, using ${\Lambda }^0=I$. All these finite-generation expectations are finite because the matrix is finite with finite entries.