11.4: A.10.3- Section 10.3 Answers

Last updated
Save as PDF

Page ID: 121461

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

2. \({\bf y'}=\left[\begin{array}{cc}{0}&{1}\\[4pt]{-\frac{P_{2}(x)}{P_{0}(x)}}&{-\frac{P_{1}(x)}{P_{0}(x)}}\end{array}\right]{\bf y}\)

3. \({\bf y'}=\left[\begin{array}{cccc}{0}&{1}&{\ldots }&{0}\\[4pt]{\vdots}&{\vdots}&{\ddots}&{\vdots}\\[4pt]{0}&{0}&{\ldots}&{1}\\[4pt]{-\frac{P_{n}(x)}{P_{0}(x)}}&{-\frac{P_{n-1}(x)}{P_{0}(x)}}&{\ldots}&{-\frac{P_{1}(x)}{P_{0}(x)}}\end{array}\right]{\bf y}\)

b. \({\bf y}=\left[\begin{array}{c}{3e^{6t}-6e^{-2t}}\\[4pt]{3e^{6t}+6e^{-2t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{2}\left[\begin{array}{cc}{e^{6t}+e^{-2t}}&{e^{6t}-e^{-2t}}\\[4pt]{e^{6t}-e^{-2t}}&{e^{6t}+e^{-2t}}\end{array}\right]{\bf k}\)

b. \({\bf y}=\left[\begin{array}{c}{6e^{-4t}+4e^{3t}}\\[4pt]{6e^{-4t}-10e^{3t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{7}\left[\begin{array}{cc}{5e^{-4t}+2e^{3t}}&{2e^{-4t}-2e^{3t}}\\[4pt]{5e^{-4t}-5e^{3t}}&{2e^{-4t}+5e^{3t}}\end{array}\right]{\bf k}\)

b. \({\bf y}=\left[\begin{array}{c}{-15e^{2t}-4e^{t}}\\[4pt]{9e^{2t}+2e^{t}}\end{array}\right]\)

c. \({\bf y}=\left[\begin{array}{cc}{-5e^{2t}+6e^{t}}&{-10e^{2t}+10e^{t}}\\[4pt]{3e^{2t}-3e^{t}}&{6e^{2t}-5e^{t}}\end{array}\right]{\bf k}\)

10.

b. \({\bf y}=\left[\begin{array}{c}{5e^{3t}-3e^{t}}\\[4pt]{5e^{3t}+3e^{t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{2}\left[\begin{array}{cc}{e^{3t}+e^{t}}&{e^{3t}-e^{t}}\\[4pt]{e^{3t}-e^{t}}&{e^{3t}+e^{t}}\end{array}\right]{\bf k}\)

11.

b. \({\bf y}=\left[\begin{array}{c}{e^{2t}-2e^{3t}+3e^{-t}}\\[4pt]{2e^{3t}-9e^{-t}}\\[4pt]{e^{2t}-2e^{3t}+21e^{-t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{6}\left[\begin{array}{ccc}{4e^{2t}+3e^{3t}-e^{-t}}&{6e^{2t}-6e^{3t}}&{2e^{2t}-3e^{3t}+e^{-t}}\\[4pt]{-3e^{3t}+3e^{-t}}&{6e^{3t}}&{3e^{3t}-3e^{-t}}\\[4pt]{4e^{2t}+3e^{3t}-7e^{-t}}&{6e^{2t}-6e^{3t}}&{2e^{2t}-3e^{3t}+7e^{-t}}\end{array}\right]{\bf k}\)

12.

b. \({\bf y}=\frac{1}{3}\left[\begin{array}{c}{-e^{-2t}+e^{4t}}\\[4pt]{-10e^{-2t}+e^{4t}}\\[4pt]{11e^{-2t}+e^{4t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{3}\left[\begin{array}{ccc}{2e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}\\[4pt]{-e^{-2t}+e^{4t}}&{2e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}\\[4pt]{-e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}&{2e^{-2t}+e^{4t}}\end{array}\right]{\bf k}\)

13.

b. \({\bf y}=\left[\begin{array}{c}{3e^{t}+3e^{-t}-e^{-2t}}\\[4pt]{3e^{t}+2e^{-2t}}\\[4pt]{-e^{-2t}}\end{array}\right]\)

c. \({\bf y}=\left[\begin{array}{ccc}{e^{-t}}&{e^{t}-e^{-t}}&{2e^{t}-3e^{-t}+e^{-2t}}\\[4pt]{0}&{e^{t}}&{2e^{t}-2e^{-2t}}\\[4pt]{0}&{0}&{e^{-2}}\end{array}\right]{\bf k}\)

14. \(YZ^{-1}\) and \(ZY^{-1}\)

Search

Text Color

Text Size

Margin Size

Font Type