Proving the relation $\det(I + xy^T ) = 1 + x^Ty$ 
Let $x$ and $y$ denote two length-$n$ column vectors. Prove that $$\det(I + xy^T ) = 1 + x^Ty$$

Is Sylvester's determinant theorem an extension of the problem? Is the approach the same?

You can also apply the property
$$\det\begin{pmatrix} A & B\\ C & D \end{pmatrix}=\det(A) \det(D-C A^{-1}B)=\det(D) \det(A-B D^{-1}C)$$
(valid when the inverses exist) to the matrix
$$\begin{pmatrix} I & -y\\ x^T & 1 \end{pmatrix}$$
(BTW: here I'm assuming column vectors; the question speaks of "column vectors", but its notation would correspond to row vectors)
