MAT244--2018F > Quiz-6

Q6 TOT 0301

(1/1)

Victor Ivrii:
Find the general solution of the given system of equations:
$$\mathbf{x}'= \begin{pmatrix} 1 &1 &2\\ 1 &2 &1\\ 2 &1 &1 \end{pmatrix}\mathbf{x}.$$

Guanyao Liang:

Boyu Zheng:
\begin{equation*}
det
\begin{pmatrix}
1-\lambda    &1           &2 \\
1         & 2-\lambda    &1 \\
2         & 1            & 1-\lambda
\end{pmatrix}
=-\lambda^3+4\lambda^2+\lambda-4=-(\lambda-1)(\lambda-4)(\lambda+1)=0
\end{equation*}
$$\lambda=1,\lambda=4,\lambda=-1$$

when $\lambda$=1
\begin{equation*}
\begin{pmatrix}
0           &1           &2 \\
1         & 1             &1 \\
2         & 1            & 0
\end{pmatrix}
\sim
\begin{pmatrix}
2           &1           &0 \\
0         & 1             &2 \\
1         & 1            & 1
\end{pmatrix}
\sim
\begin{pmatrix}
2           &0           &-2 \\
-1         & 0             &1 \\
1         & 1            & 1
\end{pmatrix}
\sim
\begin{pmatrix}
2           &0           &-2 \\
0         & 0             &0 \\
1         & 1            & 1
\end{pmatrix}
\sim
\begin{pmatrix}
2           &0           &-2 \\
1         & 1             &1 \\
0         & 0            & 0
\end{pmatrix}
\sim
\begin{pmatrix}
x_1            \\       x_2       \\       x_3
\end{pmatrix}=0
\end{equation*}
$$\text{let } x_3=t,x_1=t,x_2=-2t x= \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}t$$

when $\lambda$=4
\begin{equation*}
\begin{pmatrix}
-3           &1           &2 \\
1         & -2             &1 \\
2         & 1            & -3
\end{pmatrix}
\sim
\begin{pmatrix}
-3           &1           &2 \\
0         & -5             &5 \\
2         & 1            & -3
\end{pmatrix}
\begin{pmatrix}
x_1    \\ x_2 \\ x_3
\end{pmatrix}
=0
\end{equation*}
$$x=\begin{pmatrix} 1 \\1 \\1 \end{pmatrix}t$$
when $\lambda$=-1
\begin{equation*}
\begin{pmatrix}
2           &1           &2 \\
1         & 3             &1 \\
2         & 1            & 2
\end{pmatrix}
\sim
\begin{pmatrix}
2           &1           &2 \\
0         & 5             &0 \\
0         & 0            & 0
\end{pmatrix}
\begin{pmatrix}
x_1    \\ x_2 \\ x_3
\end{pmatrix}
=0
\end{equation*}
$$x=\begin{pmatrix}-1\\0\\1\end{pmatrix}$$
$$x(t)=c_1e^4t\begin{pmatrix}1\\1\\1\end{pmatrix}+c_2e^{-t}\begin{pmatrix}-1\\0\\1\end{pmatrix}+c_3e^t\begin{pmatrix}1\\-2\\1\end{pmatrix}$$

Chonghan Ma:
Sometimes we do not have to compute all the ref or rref. For example, when λ= 4, it is easy to observe that the sum of three columns of the matrix is 0. Sometimes it saves your time during the quiz if you can observe it directly.

Mengfan Zhu:
Let's solve this question step by step!!!