What is the inverse of the 3x3 matrix
$(\vec{a} \vec{a}+cI)^{-1}$
Where $\vec{a} \vec{a} $ is a dyad and $I$ is the identity matrix, $c$ is constant, $\vec{a}=(a_1,a_2,a_3)$ is a 3x1 vector
$\vec{a} \vec{a}+cI= \begin{pmatrix} a_1a_1& a_2a_1& a_3a_1\\ a_1a_2 & a_2a_2 & a_3a_2 \\ a_1a_3& a_2a_3& a_3a_3 \\ \end{pmatrix} + c \begin{pmatrix} 1& 0& 0\\ 0& 1 &0\\ 0& 0& 1\\ \end{pmatrix} $
This is a special case of the Sherman-Morrison Formula https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula. $$(a a^T + c I)^{-1} = \frac{1}{c}I - \frac{\frac{1}{c}Iaa^T\frac{1}{c}I}{1+\frac{1}{c}a^TIa}= \frac{1}{c}I -\frac{1}{c^2+c||a||^2}aa^T$$ Note that $c+||a||^2\neq 0$ and $c\neq0$, otherwise the identity does not hold (you get a zero in the denominator).