91Ó°ÊÓ

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

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

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

Show that \(A\) and \(B\) are not similar matrices. $$A=\left[\begin{array}{lll} 2 & 1 & 4 \\ 0 & 2 & 3 \\ 0 & 0 & 4 \end{array}\right], B=\left[\begin{array}{rrr} 1 & 0 & 0 \\ -1 & 4 & 0 \\ 2 & 3 & 4 \end{array}\right]$$

In Exercises \(I-6,\) show that \(\mathbf{v}\) is an eigenvector of \(A\) and find the corresponding eigenvalue. $$A=\left[\begin{array}{rr} -1 & 1 \\ 6 & 0 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right]$$

Show that \(A\) and \(B\) are not similar matrices. $$A=\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 1 & 1 & 4 \end{array}\right], B=\left[\begin{array}{lll} 1 & 0 & 3 \\ 1 & 2 & 2 \\ 1 & 0 & 3 \end{array}\right]$$

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. \(A=\left[\begin{array}{lll}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right]\)

Which of the stochastic matrices are regular? $$\left[\begin{array}{lll} \frac{1}{2} & 0 & 1 \\ \frac{1}{2} & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$$

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. $$A=\left[\begin{array}{ll} 1.5 & 0.5 \\ 2.0 & 3.0 \end{array}\right], \mathbf{x}_{5}=\left[\begin{array}{r} 60.625 \\ 239.500 \end{array}\right]$$

Compute the determinants using cofactor expansion along the first row and along the first column. \(\left|\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1\end{array}\right|\)