91Ó°ÊÓ

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

Find an orthogonal matrix \(P\) such that \(P^{T} A P\) diagonalizes \(A .\) Verify that \(P^{T} A P\) gives the proper diagonal form. $$A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]$$

For each matrix \(A\), 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 diagonal. \(A=\left[\begin{array}{ll}1 & -\frac{3}{2} \\ \frac{1}{2} & -1\end{array}\right]\) (See Exercise 17 Section \(7.1 .)\)

Solve the system of first-order linear differential equations. $$\begin{array}{l} y_{1}^{\prime}=-3 y_{2}+5 y_{3} \\ y_{2}^{\prime}=-4 y_{1}+4 y_{2}-10 y_{3}\\\ y_{3}^{\prime}=\quad \quad\quad\quad\quad\quad4 y_{3} \end{array}$$

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

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{rrr} 3 & 2 & -3 \\ -3 & -4 & 9 \\ -1 & -2 & 5 \end{array}\right]$$

Find an orthogonal matrix \(P\) such that \(P^{T} A P\) diagonalizes \(A .\) Verify that \(P^{T} A P\) gives the proper diagonal form. $$A=\left[\begin{array}{ll} 4 & 2 \\ 2 & 4 \end{array}\right]$$

For each matrix \(A\), 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 diagonal. \(A=\left[\begin{array}{rrr}2 & -2 & 3 \\ 0 & 3 & -2 \\ 0 & -1 & 2\end{array}\right]\) (See Exercise 21 Section \(7.1 .)\)

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{rrr} 0 & -3 & 5 \\ -4 & 4 & -10 \\ 0 & 0 & 4 \end{array}\right]$$

Solve the system of first-order linear differential equations. $$\begin{array}{lr} y_{1}^{\prime}=y_{1}-2 y_{2}+y_{3} \\ y_{2}^{\prime}= \quad\quad2 y_{2}+4 y_{3} \\ y_{3}^{\prime}\quad\quad\quad\quad\quad= 3 y_{3} \end{array}$$