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Problem 301

Given $\mathrm{A}(\mathrm{t})=\begin{array}{rr}\mid \mathrm{t}^{2} & \cos \mathrm{t} \mid \\ \mid \mathrm{e}^{\mathrm{t}} & \sin \mathrm{t} \mid\end{array}$ Find \((\mathrm{d} \mathrm{A} / \mathrm{dt})\)

Expert verified

The short answer to the question is:
\(\frac{dA}{dt} = \begin{pmatrix} 2t & -\sin(t) \\ e^t & \cos(t) \end{pmatrix}\)

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Chapter 13

Show that $\mathrm{e}^{\mathrm{At}} \mathrm{e}^{\mathrm{Bt}}=\mathrm{e}^{(\mathrm{A}+\mathrm{B}) \mathrm{t}}$ if, and only if, the matrices \(\mathrm{A}\) and B commute, i.e., \(A B=B A\).

Chapter 13

Give the matrix \(\mathrm{A}=|0 \quad 1|\) \(18 \quad-2 \mid\) find \(\mathrm{e}^{\mathrm{At}}\)

Chapter 13

\begin{tabular}{l} Given $\mathrm{A}(\mathrm{t})=\begin{array}{lcc}1 & \sin t & \cos t & t \mid \\\ & \mid(\sin t / t) & e^{t} & t^{2} \mid \\ & \mid 1 & 0 & t^{3}\end{array} \quad(t \neq 0)$ \\ \hline \end{tabular} Find \((\mathrm{d} \mathrm{A} / \mathrm{dt})\).

Chapter 13

a) If $\quad \mathrm{S}_{\mathrm{k}}=\begin{array}{cc}11 / \mathrm{k} & 1-1 / \mathrm{k}^{2} \mid \mathrm{k}=1,2, \ldots \\ 12 & 1+1 / \mathrm{k} \mid\end{array}$ find \(\lim _{\mathrm{k} \rightarrow \infty} \mathrm{S}_{\mathrm{k}}\) b) If \(\quad A_{K}=\mid 1 / k !\) \(01, \mathrm{k}=0,1, \ldots\) \(\mid 1 / 2^{\mathrm{k}} \quad\) ? find \(^{\infty} \Sigma_{\mathrm{k}=0} \mathrm{~A}_{\mathrm{k}}\) -

Chapter 13

Find \(\mathrm{f}(\mathrm{A})\) where \(\mathrm{A}=|1 \quad-2|\) $$ 14 $$ and \(\mathrm{f}(\mathrm{t})=\mathrm{t}^{2}-3 \mathrm{t}+7 .\)

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