In the following paper:

Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations

on page 79, Golub et al. have the following set of equations:

$f(\lambda) = \text{det}(I_r - U^T(\lambda - \Lambda)^{-1}U) \quad ... (1) \\ \quad =1 + \sum_{k=1}^r (-1)^k \sum_{\tau_k=1}^r\sum_{j_k=1}^n\frac{(\text{det}U_{j_1,...,j_k;\tau_1,...,\tau_k})^2}{(\lambda - \lambda_{j_1})...(\lambda - \lambda_{j_k})} \quad ... (2)$

where,

$U_{j_1,...,j_k;\tau_1,...,\tau_k} = \begin{bmatrix}e^T_{j_1}\\ \vdots\\e^T_{j_k}\end{bmatrix} U \begin{bmatrix}e_{\tau_1}&\ldots& e_{\tau_k}\end{bmatrix}$,

and $\Lambda = \text{diag}(\lambda_1,...\lambda_n)$ are the eigenvalues for a matrix $A = Q\Lambda Q^T \in \mathbb{R}^{n \times n}$ and $V=QU$.

Moreover although not explicitly stated in the paper, we infer that $\begin{bmatrix}e^T_{j_1}& \ldots &e^T_{j_k}\end{bmatrix}^T \in \mathbb{R}^{n\times k}$ and $\begin{bmatrix}e_{\tau_1}&\ldots& e_{\tau_k}\end{bmatrix} \in \mathbb{R}^{r\times k}$ since, $U \in\mathbb{R}^{n \times r}$ so that the units vectors, $e_{j_k}$ live in $n$ dimensions, and $e_{\tau_k}$ likes in $r$ dimensions

However I am not sure how he goes from one line to the other (i.e. from Equation (1) to Equation (2)).

Golub et al. state that is fairly trivial and it exploits the multi-linearity property of the determinant, however I cannot seem formulate the steps.

Let me know if you require any further information.