Following a book, I see a statement that reads
we find that (18.7.11) is euqivalent to $$ l^k\left|u_{ij}^W-\frac{b_ib_j}{l} \right|=0 $$ which can be written as $$ l^{k-1}\left|u_{ij}^W\right|\cdot\left[l - \sum_{i,j=1}^k u_W^{ij}b_ib_j\right]=0 $$
To translate, the book says that given that $0\neq l\in\mathbb{R},\;{\bf b}\in \mathbb{R}^k$ and $0<{\bf U} \in M_k(\mathbb{R})$, then $$ \det\left({\bf U} - \frac{{\bf b} {\bf b}^\top}{l} \right) = \det\left({\bf U}\right)\left[1 - \frac{{\bf b}^\top{\bf U}^{-1}{\bf b} }{l} \right]$$ Can anyone help me understand why that's true?
