# 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

Title determinants of some matrices of special form DeterminantsOfSomeMatricesOfSpecialForm 2013-03-22 14:03:41 2013-03-22 14:03:41 bwebste (988) bwebste (988) 5 bwebste (988) Result msc 15A15 BlockDeterminants