91Ó°ÊÓ

Diagonalizing a Matrix Exercises \(7-14,\) find (if possible a nonsingular matrix \(P\) such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal.. $$ A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 4 & 4 & 0 \\ 0 & 4 & 4 \end{array}\right] $$

Solve the system of first-order linear differential equations. \(y_{1}^{\prime}=-y_{1}\) \(y_{2}^{\prime}=6 y_{2}\) \(y_{3}^{\prime}=y_{3}\)

In Exercises \(7-18\), find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{cccc} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 5 \\ 0 & 0 & 5 & 3 \end{array}\right]$$

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

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{ll} 0 & 0 \\ 5 & 0 \end{array}\right] $$

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

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

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

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{rr}1 & -4 \\\\-2 & 8\end{array}\right]$$

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