Let $v$ be a (fixed) unit vector in $\mathbb { R } ^ { 3 }$. (We think of elements of $\mathbb { R } ^ { n }$ as column vectors.) Let $M = I _ { 3 } - 2 v v ^ { t }$. Pick the correct statement(s) from below. (A) $O$ is an eigenvalue of $M$. (B) $M ^ { 2 } = I _ { 3 }$. (C) 1 is an eigenvalue of $M$. (D) The eigenspace for the eigenvalue $-1$ is 2-dimensional.
Let $A \in \mathrm { GL } ( 3 , \mathbb { Q } )$ with $A ^ { t } A = I _ { 3 }$. Assume that $$A \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right]$$ for some $\lambda \in \mathbb { C }$. (A) Determine the possible values of $\lambda$. (B) Determine $x + y + z$ where $x , y , z$ is given by $$\left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = A \left[ \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right]$$