determinants of some matrices of special form

Suppose $A$ is $n\\times n$ square matrix, $u, v$ are two column $n$ -vectors, and $\\alpha$ is a scalar. Then

	$\\displaystyle\\det(A+uv^{\\operatorname{T}})$	$\\displaystyle=$	$\\displaystyle\\det A+v^{\\operatorname{T}}\\operatorname{adj}A\\,\\,u,$
	$\\displaystyle\\det\\begin{pmatrix}A&u\\\\v^{\\operatorname{T}}&\\alpha\\end{pmatrix}$	$\\displaystyle=$	$\\displaystyle\\alpha\\det A-v^{\\operatorname{T}}\\operatorname{adj}A\\,\\,u,$

where $\\operatorname{adj}A$ is the adjugate of $A$ .

References

1 V.V. Prasolov,Problems and Theorems in Linear Algebra,American Mathematical Society, 1994.