91Ó°ÊÓ

A matrix \(A\) is given along with an iterate \(\mathbf{x}_{5},\) produced as in Example 4.30 (a) Use these data to approximate a dominant eigenvector whose first component is 1 and a corresponding dominant eigenvalue. (Use three-decimal-place accuracy.) (b) Compare your approximate eigenvalue in part (a) with the actual dominant eigenvalue. $$A=\left[\begin{array}{ll}1 & 2 \\ 5 & 4\end{array}\right], \mathbf{x}_{5}=\left[\begin{array}{r}4443 \\ 11109\end{array}\right]$$

Compute the determinants in Exercises \(1-6\) using cofactor expansion along the first row and along the first column. $$\left|\begin{array}{lll} 1 & 0 & 3 \\ 5 & 1 & 1 \\ 0 & 1 & 2 \end{array}\right|$$

Show that \(A\) and \(B\) are not similar matrices. $$A=\left[\begin{array}{ll} 4 & 1 \\ 3 & 1 \end{array}\right], B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$$

Show that \(\mathbf{v}\) is an eigenvector of A and find the corresponding eigenvalue. $$A=\left[\begin{array}{ll} 0 & 3 \\ 3 & 0 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

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. $$A=\left[\begin{array}{rr} 1 & 3 \\ -2 & 6 \end{array}\right]$$

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

Show that \(\mathbf{v}\) is an eigenvector of A and find the corresponding eigenvalue. $$A=\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} 3 \\ -3 \end{array}\right]$$

A matrix \(A\) is given along with an iterate \(\mathbf{x}_{5},\) produced as in Example 4.30 (a) Use these data to approximate a dominant eigenvector whose first component is 1 and a corresponding dominant eigenvalue. (Use three-decimal-place accuracy.) (b) Compare your approximate eigenvalue in part (a) with the actual dominant eigenvalue. $$A=\left[\begin{array}{rr}7 & 4 \\ -3 & -1\end{array}\right], \mathbf{x}_{5}=\left[\begin{array}{r}7811 \\ -3904\end{array}\right]$$

Show that \(A\) and \(B\) are not similar matrices. $$A=\left[\begin{array}{rr} 3 & -1 \\ -5 & 7 \end{array}\right], B=\left[\begin{array}{rr} 2 & 1 \\ -4 & 6 \end{array}\right]$$

Compute the determinants in Exercises \(1-6\) using cofactor expansion along the first row and along the first column. $$\left|\begin{array}{rrr} 0 & 1 & -1 \\ 2 & 3 & -2 \\ -1 & 3 & 0 \end{array}\right|$$