91Ó°ÊÓ

15\. $$e^{t} \left[ \begin{array}{l}{1} \\ {5}\end{array}\right], \quad e^{t} \left[ \begin{array}{r}{-3} \\ {-15}\end{array}\right]$$

\(\mathbf{A}=\left[ \begin{array}{rrr}{-7} & {0} & {6} \\ {0} & {5} & {0} \\\ {6} & {0} & {2}\end{array}\right]\)

Let $$\mathbf{A}=\left[ \begin{array}{rrr}{2} & {-1} & {1} \\ {-1} & {2} & {1} \\ {1} & {1} & {2}\end{array}\right].$$ (a) Show that \(\mathbf{A}\) is singular. (b) Show that \(\mathbf{A} \mathbf{x}=\left[ \begin{array}{c}{3} \\ {1} \\\ {3}\end{array}\right]\) has no solutions. (c) Show that \(\mathbf{A x}=\left[ \begin{array}{c}{3} \\ {0} \\\ {3}\end{array}\right]\) has infinitely many solutions.

16\. $$\left[ \begin{array}{l}{\sin t} \\ {\cos t}\end{array}\right], \quad \left[ \begin{array}{l}{\sin 2 t} \\ {\cos 2 t}\end{array}\right]$$

\(\mathbf{A}=\left[ \begin{array}{rrr}{0} & {1} & {0} \\ {0} & {0} & {1} \\\ {-2} & {-5} & {-4}\end{array}\right]\)

17\. $$e^{2 t} \left[ \begin{array}{l}{1} \\ {0} \\ {5}\end{array}\right], \quad e^{2 t} \left[ \begin{array}{r}{1} \\ {1} \\ {-1}\end{array}\right], \quad e^{3 t} \left[ \begin{array}{l}{0} \\ {1} \\ {0}\end{array}\right]$$

$$ t \mathbf{x}^{\prime}(t)=\left[ \begin{array}{rrr}{-1} & {-1} & {0} \\ {2} & {-1} & {1} \\ {0} & {1} & {-1}\end{array}\right] \mathbf{x}(t) $$

Consider the system \(\mathbf{x}^{\prime}(t)=\mathbf{A} \mathbf{x}(t), t \geq 0,\) with $$ \mathbf{A}=\left[ \begin{array}{rr}{1} & {\sqrt{3}} \\ {\sqrt{3}} & {-1}\end{array}\right] $$ (a) Show that the matrix \(A\) has eigenvalues \(r_{1}=2\) and \(r_{2}=-2\) with corresponding eigenvectors \(\mathbf{u}_{1}=\operatorname{col}(\sqrt{3}, 1)\) and \(\mathbf{u}_{2}=\operatorname{col}(1,-\sqrt{3})\) (b) Sketch the trajectory of the solution having initial vector \(\mathbf{x}(0)=-\mathbf{u}_{1} .\) (c) Sketch the trajectory of the solution having initial vector \(\mathbf{x}(0)=\mathbf{u}_{2}\) (d) Sketch the trajectory of the solution having initial \(\quad\) vector \(\mathbf{x}(0)=\mathbf{u}_{2}-\mathbf{u}_{1}\)

\(\mathbf{A}=\left[ \begin{array}{lll}{0} & {1} & {1} \\ {1} & {0} & {1} \\\ {1} & {1} & {0}\end{array}\right], \quad \mathbf{f}(t)=\left[ \begin{array}{c}{3 e^{t}} \\ {-e^{t}} \\ {-e^{t}}\end{array}\right]\)

\(\mathbf{X}(t)=\left[ \begin{array}{ll}{\sin 2 t} & {\cos 2 t} \\ {2 \cos 2 t} & {-2 \sin 2 t}\end{array}\right]\)