91Ó°ÊÓ

Find all \(2 \times 2\) matrices for which \(\vec{e}_{1}=\left[\begin{array}{l}1 \\\ 0\end{array}\right]\) is an eigenvector with associated eigenvalue 5

For each of the matrices find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology. $$\left[\begin{array}{ccc} -1 & -1 & -1 \\ -1 & -1 & -1 \\ -1 & -1 & -1 \end{array}\right]$$

Use de Moivre's formula to express \(\cos (3 \theta)\) and \(\sin (3 \theta)\) in terms of \(\cos \theta\) and \(\sin \theta.\)

For the matrices \(A\) find closed formulas for \(A^{t},\) where \(t\) is an arbitrary positive integer. Follow the strategy outlined in Theorem 7.4 .2 and illustrated in Example \(2 .\) In Exercises 9 though \(12,\) feel free to use technology. $$A=\left[\begin{array}{rrr} 0 & 0 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & 1 \end{array}\right]$$

For each of the matrices find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology. $$\left[\begin{array}{rrr} 3 & -2 & 5 \\ 1 & 0 & 7 \\ 0 & 0 & 2 \end{array}\right]$$

For each of the matrices \(A\) find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize \(A,\) if you can. Do not use technology. $$\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$$

Consider the complex number \(z=0.8-0.7 i .\) Represent the powers \(z^{2}, z^{3}, \ldots\) in the complex plane and explain their long-term behavior.

For the matrices \(A\) in Exercises 1 through \(10,\) determine whether the zero state is a stable equilibrium of the dynamical system \(\vec{x}(t+1)=A \vec{x}(t)\). $$A=\left[\begin{array}{rrr} 0.8 & 0 & -0.6 \\ 0 & 0.7 & 0 \\ 0.6 & 0 & 0.8 \end{array}\right]$$

For the matrices \(A\) find closed formulas for \(A^{t},\) where \(t\) is an arbitrary positive integer. Follow the strategy outlined in Theorem 7.4 .2 and illustrated in Example \(2 .\) In Exercises 9 though \(12,\) feel free to use technology. $$A=\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 2 \end{array}\right]$$

For each of the matrices \(A\) find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize \(A,\) if you can. Do not use technology. $$\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$$