Suppose $\hat{x}\in\mathbb{R}^d$ is a normalized vector and $\alpha\in(0, 1)$ a scalar. Is there a way to compute the determinant of the following matrix quickly? $$ I_d - \alpha\hat{x}\hat{x}^\top $$ To me this structure seems to have a lot of structure so I'd expect some trick to do the job.
This question aims to create an &quot;&quot;  of numerous questions that ask about determinants of specific matrices (I may have missed a few):              The general question of this type is  Let $A$ be a square matrix of rank$~1$, let $I$ the identity matrix of the same size, and $\lambda$ a scalar. What is the determinant of $A+\lambda I$?  A clearly very closely related question is  What is the characteristic polynomial of a matrix $A$ of rank$~1$?
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