91Ó°ÊÓ

Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and \((d)\) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 \\ -2 & 1 & 2 & -1 \end{array}\right]$$

Show that \(\lambda\) is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. $$A=\left[\begin{array}{rrr} 1 & 0 & 2 \\ -1 & 1 & 1 \\ 2 & 0 & 1 \end{array}\right], \lambda=-1$$

Use the power method to approximate the dominant eigenvalue and eigenvector of \(A .\) Use the given initial vector \(\mathbf{x}_{0},\) the specified number of iterations \(k,\) and three-decimal-place accuracy. $$A=\left[\begin{array}{ll}7 & 2 \\ 2 & 3\end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 0\end{array}\right], k=6$$

Use the power method to approximate the dominant eigenvalue and eigenvector of \(A .\) Use the given initial vector \(\mathbf{x}_{0},\) the specified number of iterations \(k,\) and three-decimal-place accuracy. $$A=\left[\begin{array}{lr}3.5 & 1.5 \\ 1.5 & -0.5\end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 0\end{array}\right], k=6$$

Show that \(\lambda\) is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. $$A=\left[\begin{array}{rrr} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 4 & 2 & 0 \end{array}\right], \lambda=2$$

Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and \((d)\) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{llll} 4 & 0 & 1 & 0 \\ 0 & 4 & 1 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 0 \end{array}\right]$$

Compute the determinants in using cofactor expansion along any row or column that seems convenient $$\left|\begin{array}{lll} 0 & a & 0 \\ b & c & d \\ 0 & e & 0 \end{array}\right|$$

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & 0 & 1 \end{array}\right]$$

Compute the determinants in using cofactor expansion along any row or column that seems convenient $$\left|\begin{array}{rrrr} 1 & -1 & 0 & 3 \\ 2 & 5 & 2 & 6 \\ 0 & 1 & 0 & 0 \\ 1 & 4 & 2 & 1 \end{array}\right|$$

Use the power method to approximate the dominant eigenvalue and eigenvector of \(A .\) Use the given initial vector \(\mathbf{x}_{0},\) the specified number of iterations \(k,\) and three-decimal-place accuracy. $$A=\left[\begin{array}{lll}9 & 4 & 8 \\ 4 & 15 & -4 \\ 8 & -4 & 9\end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right], k=5$$