A.10.2 Section 10.2 Answers

1.

1. \(\bf{y'}=\left[\begin{array}{cc}{2}&{4}\\[4pt]{4}&{2}\end{array}\right]{\bf y}\)
2. \(\bf{y'}=\left[\begin{array}{cc}{-2}&{-2}\\[4pt]{-5}&{1}\end{array}\right]{\bf y}\)
3. \(\bf{y'}=\left[\begin{array}{cc}{-4}&{-10}\\[4pt]{3}&{7}\end{array}\right]{\bf y}\)
4. \(\bf{y'}=\left[\begin{array}{cc}{2}&{1}\\[4pt]{1}&{2}\end{array}\right]{\bf y}\)

2.

1. \(\bf{y'}=\left[\begin{array}{ccc}{-1}&{2}&{3}\\[4pt]{0}&{1}&{6}\\[4pt]{0}&{0}&{-2}\end{array}\right]{\bf y}\)
2. \(\bf{y'}=\left[\begin{array}{ccc}{0}&{2}&{2}\\[4pt]{2}&{0}&{2}\\[4pt]{2}&{2}&{0}\end{array}\right]{\bf y}\)
3. \(\bf{y'}=\left[\begin{array}{ccc}{-1}&{2}&{2}\\[4pt]{2}&{-1}&{2}\\[4pt]{2}&{2}&{-1}\end{array}\right]{\bf y}\)
4. \(\bf{y'}=\left[\begin{array}{ccc}{3}&{-1}&{-1}\\[4pt]{-2}&{3}&{2}\\[4pt]{4}&{-1}&{-2}\end{array}\right]{\bf y}\)

3.

1. \(\bf{y'}=\left[\begin{array}{cc}{1}&{1}\\[4pt]{-2}&{4}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{1}\\[4pt]{0}\end{array}\right]\)
2. \(\bf{y'}=\left[\begin{array}{cc}{5}&{3}\\[4pt]{-1}&{1}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{9}\\[4pt]{-5}\end{array}\right]\)

4.

1. \(\bf{y'}=\left[\begin{array}{ccc}{6}&{4}&{4}\\[4pt]{-7}&{-2}&{-1}\\[4pt]{7}&{4}&{3}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{3}\\[4pt]{-6}\\[4pt]{4}\end{array}\right]\)
2. \(\bf{y'}=\left[\begin{array}{ccc}{8}&{7}&{7}\\[4pt]{-5}&{-6}&{-9}\\[4pt]{5}&{7}&{10}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{2}\\[4pt]{-4}\\[4pt]{3}\end{array}\right]\)

5.

1. \({\bf y'}=\left[\begin{array}{cc}{-3}&{2}\\[4pt]{-5}&{3}\end{array}\right] {\bf y}+ \left[\begin{array}{c}{3-2t}\\[4pt]{6-3t}\end{array}\right]\)
2. \({\bf y'}=\left[\begin{array}{cc}{3}&{1}\\[4pt]{-1}&{1}\end{array}\right]{\bf y} + \left[\begin{array}{c}{-5e^{t}}\\[4pt]{e^{t}}\end{array}\right]\)

10.

1. \(\frac{d}{dt}Y^{2}=Y'Y+YY'\)
2. \(\frac{d}{dt}Y^{n}=Y'Y^{n-1}+YY'Y^{n-2}+Y^{2}Y'Y^{n-3}+\ldots + Y^{n-1}Y'=\sum_{r=0}^{n-1}Y^{r}Y'Y^{n-r-1}\)

13. \(B=(P'+PA)P^{-1}\)

---