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

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

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

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

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

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

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

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

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

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