91Ó°ÊÓ

Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by Theorem \(7.6 .\)) $$\left[\begin{array}{ll} 2 & 0 \\ 5 & 2 \end{array}\right]$$

Solve the system of first-order linear differential equations. $$\begin{array}{l} y_{1}^{\prime}=-y_{1} \\ y_{2}^{\prime}=-2 y_{2} \\ y_{3}^{\prime}=y_{3} \end{array}$$

Determine whether the matrix is orthogonal. $$\left[\begin{array}{rrr} -\frac{4}{5} & 0 & \frac{3}{5} \\ 0 & 1 & 0 \\ \frac{3}{5} & 0 & \frac{4}{5} \end{array}\right]$$

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

Solve the system of first-order linear differential equations. $$\begin{array}{l} y_{1}^{\prime}=y_{1}-4 y_{2} \\ y_{2}^{\prime}=2 y_{2} \end{array}$$

Determine whether the matrix is orthogonal. $$\left[\begin{array}{ccc} \frac{\sqrt{2}}{2} & -\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{6}}{3} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{6}}{6} & -\frac{\sqrt{3}}{3} \end{array}\right]$$

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{lll} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \end{array}\right]$$

Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by Theorem \(7.6 .\)) $$\left[\begin{array}{rrr} 3 & 2 & -3 \\ -3 & -4 & 9 \\ -1 & -2 & 5 \end{array}\right]$$

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{rrr} -5 & 0 & 0 \\ 3 & 7 & 0 \\ 4 & -2 & 3 \end{array}\right]$$

Solve the system of first-order linear differential equations. $$\begin{array}{lr} y_{1}^{\prime}= & y_{1}-4 y_{2} \\ y_{2}^{\prime}= & -2 y_{1}+8 y_{2} \end{array}$$