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 $$\left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = A \left[ \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right]$$
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
$$\left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = A \left[ \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right]$$