> 
> I am asked to prove
> 
> 
> $$\mathbf{\det(I+xy^T+uv^T)}=(1+\mathbf{y^Tx})(1+\mathbf{v^Tu)-(x^Tv)(y^Tu)}$$
> 
> 
> By first proving that
> $\mathbf{\det(I+xy^T)}=1+\mathbf{y^Tx}$, where $\mathbf{x}$ and $\mathbf{y}$ are $n$ vector.
> 
> 
> 

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Assuming that $x\neq0$, we can find vectors $w\_1, w\_2, \cdots, w\_{n-1}$ such that the matrix $Q$ defined by $$Q=[x,w\_1,w\_2,\cdots, w\_{n-1}]$$ is nonsingular and $x=Qe\_1$, where $e\_1=(1,0,0,\cdots,0)^T$. If we define $$y^TQ=(z\_1,z\_2,\cdots,z\_n),$$ then $$z\_1=y^TQe\_1=y^TQ(Q^{-1}x)=y^Tx,$$ and $$\det(I+xy^T)=\det(Q^{-1}(I+xy^T)Q)=\det(I+e\_1y^TQ).$$

After here I am not quite sure how to close the argument and then go on to prove that $\mathbf{\det(I+xy^T+uv^T)}=(1+\mathbf{y^Tx})(1+\mathbf{v^Tu)-(x^Tv)(y^Tu)}$.