Question

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 \(\operatorname{det}(A+I)=1+\operatorname{det} A\).