91Ó°ÊÓ

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

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

A matrix \(A\) is given along with an iterate \(\mathbf{x}_{k},\) produced using the power method, as in Example 4.31 (a) Approximate the dominant eigenvalue and eigenvector by computing the corresponding \(m_{k}\) and \(\mathbf{y}_{k}\). (b) Verify that you have approximated an eigenvalue and an eigenvector of \(A\) by comparing \(A \mathbf{y}_{k}\) with \(m_{k} \mathbf{y}_{k}.\) $$A=\left[\begin{array}{rrr}4 & 0 & 6 \\ -1 & 3 & 1 \\ 6 & 0 & 4\end{array}\right], \mathbf{x}_{8}=\left[\begin{array}{r}10.000 \\ 0.001 \\\ 10.000\end{array}\right]$$

A matrix \(A\) is given along with an iterate \(\mathbf{x}_{k},\) produced using the power method, as in Example 4.31 (a) Approximate the dominant eigenvalue and eigenvector by computing the corresponding \(m_{k}\) and \(\mathbf{y}_{k}\). (b) Verify that you have approximated an eigenvalue and an eigenvector of \(A\) by comparing \(A \mathbf{y}_{k}\) with \(m_{k} \mathbf{y}_{k}.\) $$A=\left[\begin{array}{rrr}1 & 2 & -2 \\ 1 & 1 & -3 \\ 0 & -1 & 1\end{array}\right], \mathbf{x}_{10}=\left[\begin{array}{r}3.415 \\ 2.914 \\\ -1.207\end{array}\right]$$

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

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

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{ll} 5 & 2 \\ 2 & 5 \end{array}\right]$$

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

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. $$A=\left[\begin{array}{rr}14 & 12 \\ 5 & 3\end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right], k=5$$

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{ll} -3 & 4 \\ -1 & 1 \end{array}\right]$$