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Let $A$ be the rank one $n \times n$ matrix $A=\left(v_{i} v_{j}\right)$, where $\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)$ is a non-zero real vector.

a) Find its eigenvalues and eigenvectors.
b) Find the eigenvalues and eigenvectors for $A+c I$, where $c \in \mathbb{R}$.
c) Find a formula for $(I+A)^{-1}$. [ANSWER: $(I+A)^{-1}=I-\frac{1}{1+\|\vec{v}\|^{2}} A$.]
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