Diagonalise: a) \( A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right] \) colculate \( A^{3} \), using \( A^{n}=P D^{n} P^{-1} \)

Hausaufgabe 33: (Eigenwertproblem und Exponentialmatrix) \( \quad(3+1+2=6 \) Punkte) Zur Behandlung des Spins in der Quantenmechanik benötigt man die sogenannte Pauli-Matrix \[ \sigma=\left(\begin{array}{rr}0 & -i \\ i & 0\end{array}\right) \] a) Bestimmen Sie ihre Eigenwerte \( \lambda_{1} \) und \( \lambda_{2} \) sowie die zugehörigen Eigenvektoren \( \vec{v}_{1} \) und \( \vec{v}_{2} \). b) Verifizieren Sie für die von Ihnen gefundenen Eigenwerte und Eigenvektoren jeweils die Gleichung \( \sigma \vec{v}_{i}=\lambda_{i} \vec{v}_{i} \). c) Berechnen Sie die Exponentialmatrix \( e^{i \varphi \sigma} \) für \( \varphi \in \mathbb{R} \).

Simplify \( z=\frac{(3+i)^{3}}{3-i} \) in the form \( a+i b \) where \( i=\sqrt{-1} \) and find the value of \( |z| \).

Question 67 (2 points) Presence is the subjective feeling of being in a virtual environment where the viewer is unaware of the technological medium through which the environment is being presented. True False Question 68 ( 2 points) This occurs when people are influenced or affected by their digital representations, such as avatars Immersion The SIDE Model Proteus Effect

Pablo wants to get an overview of any project he undertakes, and always asks for the "big picture." What is Pablo expressing a preference for? a. kinesthetic learning b. read/write learning c. visual learning d. auditory learning

Working on this case study, it was a challenging and rewarding experience that significantly shaped my understanding of inclusive education. Developing an effective and learner-specific instructional plan for this student required me to consider their unique strengths, interests, and areas that needed support I found myself reflecting on how important it is to tailor our teaching strategies to fit individual needs, rather than applying a one-size fits all approach. Collaborating with my peers to create an inclusive plan added an extra layer of depth to this process, as we engaged in thoughtful discussions about best practices and shared effective strategies. Through our collective efforts, we highlighted the value of teamwork in crafting an a positive social environment. This experience reinforced the notion that every student,