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.

Title	determinants of some matrices of special form
Canonical name	DeterminantsOfSomeMatricesOfSpecialForm
Date of creation	2013-03-22 14:03:41
Last modified on	2013-03-22 14:03:41
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	5
Author	bwebste (988)
Entry type	Result
Classification	msc 15A15
Related topic	BlockDeterminants