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
$det(A+u{v}^{\mathrm{T}})$ | $=$ | $detA+{v}^{\mathrm{T}}\mathrm{adj}Au,$ | ||
$det\left(\begin{array}{cc}\hfill A\hfill & \hfill u\hfill \\ \hfill {v}^{\mathrm{T}}\hfill & \hfill \alpha \hfill \end{array}\right)$ | $=$ | $\alpha detA-{v}^{\mathrm{T}}\mathrm{adj}Au,$ |
where $\mathrm{adj}A$ 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 |
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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 |