<s>[INST] 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?
 [/INST] Hint: Decomposing $$ \begin{pmatrix} 1 & -y^T\\ x & I \end{pmatrix} $$
as lower $\cdot$ upper and upper $\cdot$ lower gives
$$ \begin{pmatrix} 1 & 0\\ x & I\end{pmatrix} \cdot \begin{pmatrix} 1 & -y^T\\ 0 & I + xy^T \end{pmatrix} = \begin{pmatrix} 1 + x^Ty & -y^T\\ 0 & I\end{pmatrix} \cdot \begin{pmatrix} 1 & 0\\ x & I \end{pmatrix}. $$
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