91Ó°ÊÓ

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{ll}1 & 2 \\\2 & 1\end{array}\right]$$

In Exercises \(7-18\), find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{rrrrr} 2 & -1 & 0 & 0 & 0 \\ -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{array}\right]$$

Solve the system of first-order linear differential equations. \(y_{1}^{\prime}=-0.3 y_{1}\) \(y_{2}^{\prime}=0.4 y_{2}\) \(y_{3}^{\prime}=-0.6 y_{3}\)

Solve the system of first-order linear differential equations. \(y_{1}^{\prime}=-\frac{2}{3} y_{1}\) \(y_{2}^{\prime}=-\frac{3}{5} y_{2}\) \(y_{3}^{\prime}=-8 y_{3}\)

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{rr} 1 & 0 \\ -2 & 1 \end{array}\right] $$

In Exercises \(7-18\), find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{rrrrr} 1 & -1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & -1 & 1 \end{array}\right]$$

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{rr}-2 & 4 \\\1 & 1\end{array}\right]$$

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right] $$

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{cc}1 & -\frac{3}{2} \\\\\frac{1}{2} & -1\end{array}\right]$$

Solve the system of first-order linear differential equations. \(y_{1}^{\prime}=7 y_{1}\) \(y_{2}^{\prime}=9 y_{2}\) \(y_{3}^{\prime}=-7 y_{3}\) \(y_{4}^{\prime}=-9 y_{4}\)