Q4: Given the matrix \( A=\left(\begin{array}{ccc}7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 3\end{array}\right) \) Where \( X 1=\left(\begin{array}{c}0 \\ 0.5 \\ 1\end{array}\right) \quad, X 2=\left(\begin{array}{c}0 \\ 0.5 \\ 1\end{array}\right), X 3=\left(\begin{array}{c}1 \\ -2 \\ 1\end{array}\right) \) Find: 1) the rank of \( A \) ? 2) the eigen values and eigen vectors? 3) satisfy \( P^{\prime} / A P=\operatorname{Diag}\left(\begin{array}{lll}\lambda_{1} & \lambda_{2} & \lambda_{3}\end{array}\right) \) or \( N o \) ?

IX. Please determine whether the set of vectors \( \left\{\begin{array}{lll}\alpha_{1} & \alpha_{2} & \alpha_{3}\end{array}\right\} \) is linearly dependent or not, where \( \alpha_{1}=\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)^{T}, \alpha_{2}=\left(\begin{array}{llll}2 & 1 & 1\end{array}\right)^{T}, \alpha_{3}=\left(\begin{array}{lll}3 & 2 & 1\end{array}\right)^{T} \). (Write the answer for the questions on your answer sheet. 10 points)

Exorcise 3.11 . Let \( \mathrm{A}=\left(\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 6 & -5\end{array}\right) \). Find the matrix obtained from \( A \) by applying the following elementary row operation. \( \begin{array}{lll}\text { a. } R_{1} \leftrightarrow R_{3} & \text { b. } R_{2} \rightarrow R_{2}-4 R_{1} & \text { c. } R_{3} \rightarrow \frac{1}{3} R_{3}\end{array} \)

Calcular la proyección del vector \( \vec{u}=(-3,2,1) \) sobre el vector \( \vec{v}=(-1,2,4) \) a. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \) b. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21}, \frac{44}{21}\right) \) c. \( \vec{w}=\left(-\frac{11}{21},-\frac{22}{21}, \frac{44}{21}\right) \) d. \( \vec{w}=\left(\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \)

1. Representar en un diagrama de Venn el conjunto \( [A-(A \cap B)]^{C} \). Considere a) \( A \) subconjunto de \( B \) b) \( B \) subconjunto de \( A \)

4. If \( \quad\left[\begin{array}{cccccc}0 & 0 & -3 & -2 & 0 & 5 \\ -2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -2 & 3 & 0 & 0 & 0 \\ -2 & 8 & 3 & 6 & 2 & 7 \\ 0 & 2 & 3 & 0 & 0 & 0\end{array}\right] \operatorname{compute} \operatorname{det}(A),(\operatorname{adj}(A))^{-1} \) and \( \operatorname{det}(\operatorname{adj}(A)) \)