91Ó°ÊÓ

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

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. $$\left[\begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{array}\right]$$

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{rrrr} 1 & -3 & 3 & 3 \\ -1 & 4 & -3 & -3 \\ -2 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right] $$

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. $$\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right]$$

Determining a Sufficient Condition for Diagonalization In Exercises \(23-26,\) find the eigenvalues of the matrix and determine whether there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. $$ \left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right] $$

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{rrr}1 & 2 & -2 \\\\-2 & 5 & -2 \\\\-6 & 6 & -3\end{array}\right]$$

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

Determining a Sufficient Condition for Diagonalization In Exercises \(23-26,\) find the eigenvalues of the matrix and determine whether there is a sufficient number of eigenvalues to guarantee that the matrix is 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. \(y_{1}^{\prime}=y_{1}-y_{2}\) \(y_{2}^{\prime}=2 y_{1}+4 y_{2}\)

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. $$\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$