91Ó°ÊÓ

Show that \(\lambda\) is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. $$A=\left[\begin{array}{rr} 0 & 4 \\ -1 & 5 \end{array}\right], \lambda=1$$

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. $$A=\left[\begin{array}{rrrr} 3 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 1 & 1 \end{array}\right]$$

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

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. $$A=\left[\begin{array}{llll} 2 & 1 & 1 & 0 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right]$$

Compute the determinants in using cofactor expansion along any row or column that seems convenient $$\left|\begin{array}{ccc} \cos \theta & \sin \theta & \tan \theta \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{array}\right|$$

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

Show that \(\lambda\) is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. $$A=\left[\begin{array}{rr} 0 & 4 \\ -1 & 5 \end{array}\right], \lambda=4$$

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. $$A=\left[\begin{array}{rr}-6 & 4 \\ 8 & -2\end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 0\end{array}\right], k=6$$

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

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. $$A=\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 \\ -2 & 1 & 2 & -1 \end{array}\right]$$