The solution of $$\mathbf { X } ^ { \prime } = \left( \begin{array} { l l }
1 & - 2 \
1 & - 1
\end{array} \right) \mathbf { X }$$ is
A) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l }
2 \
1
\end{array} \right) \cos t - \left( \begin{array} { c }
0 \
- 1
\end{array} \right) \sin t \right] + c _ { 2 } \left[ \left( \begin{array} { l }
2 \
1
\end{array} \right) \sin t + \left( \begin{array} { c }
0 \
- 1
\end{array} \right) \cos t \right]$$
B) $$X = c _ { 1 } \left( \begin{array} { l }
1 \
0
\end{array} \right) e ^ { \sqrt { 3 } t } + c _ { 2 } \left( \begin{array} { l }
0 \
1
\end{array} \right) e ^ { - \sqrt { 3 } t }$$
C) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l }
1 \
0
\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l }
0 \
1
\end{array} \right) e ^ { - t }$$
D) $$\begin{array} { l }
\mathrm { X } = c _ { 1 } \left[ \left( \begin{array} { l }
2 \
1
\end{array} \right) \cos ( \sqrt { 3 } t ) - \left( \begin{array} { c }
0 \
- 1
\end{array} \right) \sin ( \sqrt { 3 } t ) \right] + \
c _ { 2 } \left[ \left( \begin{array} { l }
2 \
1
\end{array} \right) \sin ( \sqrt { 3 } t ) + \left( \begin{array} { c }
0 \
- 1
\end{array} \right) \cos ( \sqrt { 3 } t ) \right]
\end{array}$$
E) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l }
2 \
1
\end{array} \right) \cos ( 2 t ) - \left( \begin{array} { c }
0 \
- 1
\end{array} \right) \sin ( 2 t ) \right] + c _ { 2 } \left[ \left( \begin{array} { l }
2 \
1
\end{array} \right) \sin ( 2 t ) + \left( \begin{array} { c }
0 \
- 1
\end{array} \right) \cos ( 2 t ) \right]$$

Question

The solution of $$\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 
1 & - 2 \
1 & - 1
\end{array} \right) \mathbf { X }$$ is
A) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \cos t - \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \sin t \right] + c _ { 2 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \sin t + \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \cos t \right]$$
B) $$X = c _ { 1 } \left( \begin{array} { l } 
1 \
0
\end{array} \right) e ^ { \sqrt { 3 } t } + c _ { 2 } \left( \begin{array} { l } 
0 \
1
\end{array} \right) e ^ { - \sqrt { 3 } t }$$
C) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 
1 \
0
\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 
0 \
1
\end{array} \right) e ^ { - t }$$
D) $$\begin{array} { l } 
\mathrm { X } = c _ { 1 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \cos ( \sqrt { 3 } t ) - \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \sin ( \sqrt { 3 } t ) \right] + \
c _ { 2 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \sin ( \sqrt { 3 } t ) + \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \cos ( \sqrt { 3 } t ) \right]
\end{array}$$
E) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \cos ( 2 t ) - \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \sin ( 2 t ) \right] + c _ { 2 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \sin ( 2 t ) + \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \cos ( 2 t ) \right]$$

Quizplus · Accepted Answer

The Answer of The solution of $$\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 
1 & - 2 \
1 & - 1
\end{array} \right) \mathbf { X }$$ is
A) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \cos t - \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \sin t \right] + c _ { 2 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \sin t + \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \cos t \right]$$
B) $$X = c _ { 1 } \left( \begin{array} { l } 
1 \
0
\end{array} \right) e ^ { \sqrt { 3 } t } + c _ { 2 } \left( \begin{array} { l } 
0 \
1
\end{array} \right) e ^ { - \sqrt { 3 } t }$$
C) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 
1 \
0
\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 
0 \
1
\end{array} \right) e ^ { - t }$$
D) $$\begin{array} { l } 
\mathrm { X } = c _ { 1 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \cos ( \sqrt { 3 } t ) - \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \sin ( \sqrt { 3 } t ) \right] + \
c _ { 2 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \sin ( \sqrt { 3 } t ) + \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \cos ( \sqrt { 3 } t ) \right]
\end{array}$$
E) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \cos ( 2 t ) - \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \sin ( 2 t ) \right] + c _ { 2 } \left[ \left( \begin{array} { l } 
2 \
1
\end{array} \right) \sin ( 2 t ) + \left( \begin{array} { c } 
0 \
- 1
\end{array} \right) \cos ( 2 t ) \right]$$

The Solution Of X′=(1−21−1)X\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right) \mathbf { X }X′=(11​−2−1​)X

The Solution Of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right) \mathbf { X }$