91Ó°ÊÓ

Find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{rrr} 0 & 4 & 4 \\ 4 & 2 & 0 \\ 4 & 0 & -2 \end{array}\right]$$

Show that the matrix is not diagonalizable. $$\left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right]$$

Determine whether \(\mathbf{x}\) is an eigenvector of \(A\). \(A=\left[\begin{array}{rrr}-1 & -1 & 1 \\ -2 & 0 & -2 \\ 3 & -3 & 1\end{array}\right]\) (a) \(\mathbf{x}=(2,-4,6)\) (b) \(\mathbf{x}=(2,0,6)\) (c) \(\mathbf{x}=(2,2,0)\) (d) \(\mathbf{x}=(-1,0,1)\)

Find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{array}\right]$$

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

Solve the system of first-order linear differential equations. $$\begin{array}{l} y_{1}^{\prime}=-3 y_{1} \\ y_{2}^{\prime}=4 y_{2} \end{array}$$

Find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{rrr} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{array}\right]$$

Determine whether \(\mathbf{x}\) is an eigenvector of \(A\). \(A=\left[\begin{array}{rrr}1 & 0 & 5 \\ 0 & -2 & 4 \\ 1 & -2 & 9\end{array}\right]\) (a) \(\mathbf{x}=(1,1,0)\) (b) \(\mathbf{x}=(-5,2,1)\) (c) \(\mathbf{x}=(0,0,0)\) (d) \(\mathbf{x}=(2 \sqrt{6}-3,-2 \sqrt{6}+6,3)\)

Show that the matrix is not diagonalizable. $$\left[\begin{array}{rrr} 2 & 1 & -1 \\ 0 & -1 & 2 \\ 0 & 0 & -1 \end{array}\right]$$

Solve the system of first-order linear differential equations. $$\begin{array}{l} y_{1}^{\prime}=-y_{1} \\ y_{2}^{\prime}=6 y_{2} \end{array}$$