91Ó°ÊÓ

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

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{ll}\frac{1}{4} & \frac{1}{4} \\\\\frac{1}{2} & 0\end{array}\right]$$

In Exercises \(19-32,\) determine whether the matrix is orthogonal. If the matrix is orthogonal, then show that the column vectors of the matrix form an orthonormal set. $$\left[\begin{array}{rr} \frac{4}{9} & -\frac{4}{9} \\ \frac{4}{9} & \frac{3}{9} \end{array}\right]$$

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{rrr} 3 & 2 & -2 \\ 0 & -2 & 3 \\ 0 & 0 & -2 \end{array}\right] $$

Solve the system of first-order linear differential equations. \(y_{1}^{\prime}=-0.1 y_{1}\) \(y_{2}^{\prime}=-\frac{7}{4} y_{2}\) \(y_{3}^{\prime}=-2 \pi y_{3}\) \(y_{4}^{\prime}=\sqrt{5} y_{4}\)

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

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{rrr}2 & -2 & 3 \\\0 & 3 & -2 \\\0 & -1 & 2\end{array}\right]$$

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

In Exercises \(19-32,\) determine whether the matrix is orthogonal. If the matrix is orthogonal, then show that the column vectors of the matrix form an orthonormal set. $$\left[\begin{array}{rr} -0.936 & -0.352 \\ 0.352 & -0.936 \end{array}\right]$$

Solve the system of first-order linear differential equations. \(y_{1}^{\prime}=y_{1}-4 y_{2}\) \(y_{2}^{\prime}=-2 y_{1}+8 y_{2}\)