= 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 $t$ is
$$
\boxed{\mathbb E_i|Z_t|=e_i^T\Lambda^t\mathbf1=\sum_{j=1}^l(\Lambda^t)_{ij}.}
$$
This also covers $t=0$, using $\Lambda^0=I$. All these finite-generation <expectations> are finite because the <matrix> is finite with finite entries.
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