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Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix whose rank is 1 . Let $v:=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be a basis for the image of $A$.
a) Show that $a_{i j}=v_{i} w_{j}$ for some vector $w:=\left(w_{1}, \ldots, w_{n}\right) \neq 0 .$
b) If $A$ has a non-zero eigenvalue $\lambda_{1}$, show that

c) If the vector $z=\left(z_{1}, \ldots, z_{n}\right)$ satisfies $\langle z, w\rangle=0$, show that $z$ is an eigenvector with eigenvalue $\lambda=0$.
d) If trace $(A) \neq 0$, show that $\lambda=\operatorname{trace}(A)$ is an eigenvalue of $A$. What is the corresponding eigenvector?
e) If trace $(A) \neq 0$, prove that $A$ is similar to the $n \times n$ matrix

$\left(\begin{array}{cccc}c & 0 & \ldots & 0 \\ 0 & 0 & \ldots & 0 \\ \ldots \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & 0\end{array}\right)$

where $c=\operatorname{trace}(A)$
f) If $\operatorname{trace}(A)=1$, show that $A$ is a projection, that is, $A^{2}=A$.
g) What can you say if $\operatorname{trace}(A)=0$ ?
h) Show that $\operatorname{det}(A+I)=1+\operatorname{det} A$.
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