图源：Total Lunar Eclipse: 2019 January

Lunar Eclipse Sequnce over 13th Century Church
Credit: Rafael Schmall
Location: Pusztatorony, Somogyvámos, Hungary

@德洛奈曲面 这不就是李炯生的《线性代数》P71页 第二章第4节的习题1嘛……
\[
\color{purple}{
\left\{
\begin{split}
x_1&+&&x_2&+&2&x_3&+&3&x_4&=&&1\\
3x_1&-&&x_2&-&&x_3&-&2&x_4&=&-&4\\
2x_1&+&3&x_2&-&&x_3&-&&x_4&=&-&6\\
x_1&+&2&x_2&+&3&x_3&-&&x_4&=&-&4
\end{split}
\right. }
\]
\[
D_a=\left|\begin{array}{ccccccc}
1&1&2&3\\
3&-1&-1&-2\\
2&3&-1&-1\\
1&2&3&-1
\end{array}
\right|
\overset{r_2-3r_1\\r_3-2r_1\\r_4-r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
1&1&2&3\\
0&-4&-7&-11\\
0&1&-5&-7\\
0&1&1&-4
\end{array}
\right|\\
\overset{r_2\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&-4\\
0&1&-5&-7\\
0&-4&-7&-11
\end{array}
\right|
\overset{r_3-r_2\\r_4+4r_2}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&-4\\
0&0&-6&-3\\
0&0&-3&-27
\end{array}
\right|\\
\overset{r_3\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}+}\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&-4\\
0&0&-3&-27\\
0&0&-6&-3\\
\end{array}
\right|
\overset{r_4-2r_3}{\overline{\overline{\hspace{3cm}}}}
{\color{red}+}\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&-4\\
0&0&-3&-27\\
0&0&0&51\\
\end{array}
\right|={\color{blue}-153}
\]
\[
{D_a}_1=\left|\begin{array}{ccccccc}
1&1&2&3\\
-4&-1&-1&-2\\
-6&3&-1&-1\\
-4&2&3&-1
\end{array}
\right|
\overset{r_2+4r_1\\r_3+6r_1\\r_4+4r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
1&1&2&3\\
0&3&7&10\\
0&9&11&17\\
0&6&11&11
\end{array}
\right|\\
\overset{\\r_3-3r_2\\r_4-2r_2}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
1&1&2&3\\
0&3&7&10\\
0&0&-10&-13\\
0&0&-3&-9
\end{array}
\right|=
-3\left|\begin{array}{ccccccc}
1&1&2&3\\
0&3&7&10\\
0&0&-10&-13\\
0&0&1&3
\end{array}
\right|\\
\overset{r_3\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
\,{\color{red}+}3\left|\begin{array}{ccccccc}
1&1&2&3\\
0&3&7&10\\
0&0&1&3\\
0&0&-10&-13
\end{array}
\right|
\overset{r_4+10r_3}{\overline{\overline{\hspace{3cm}}}}
\,3\left|\begin{array}{ccccccc}
1&1&2&3\\
0&3&7&10\\
0&0&1&3\\
0&0&0&17
\end{array}
\right|
={\color{blue}153}
\]
\[
{D_a}_2=\left|\begin{array}{ccccccc}
1&1&2&3\\
3&-4&-1&-2\\
2&-6&-1&-1\\
1&-4&3&-1
\end{array}
\right|
\overset{r_2-3r_1\\r_3-2r_1\\r_4-r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
1&1&2&3\\
0&-7&-7&-11\\
0&-8&-5&-7\\
0&-5&1&-4
\end{array}
\right|\\
=
-7\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&\frac{11}{7}\\
0&-8&-5&-7\\
0&-5&1&-4
\end{array}
\right|
\overset{r_4+5r_2}{\overline{\overline{\hspace{3cm}}}}
-7\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&\frac{11}{7}\\
0&-8&-5&-7\\
0&0&6&\frac{27}{7}
\end{array}
\right|\\
\overset{r_3+8r_2}{\overline{\overline{\hspace{3cm}}}}
-7\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&\frac{11}{7}\\
0&0&3&\frac{39}{7}\\
0&0&6&\frac{27}{7}
\end{array}
\right|
\overset{r_4-2r_3}{\overline{\overline{\hspace{3cm}}}}
-7\left|\begin{array}{ccccccc}
1&1&2&3\\
0&1&1&\frac{11}{7}\\
0&0&3&\frac{39}{7}\\
0&0&0&-\frac{51}{7}
\end{array}
\right|
={\color{blue}153}
\]
\[
{D_a}_3=\left|\begin{array}{ccccccc}
1&1&1&3\\
3&-1&-4&-2\\
2&3&-6&-1\\
1&2&-4&-1
\end{array}
\right|
\overset{r_2-3r_1\\r_3-2r_1\\r_4-r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
1&1&1&3\\
0&-4&-7&-11\\
0&1&-8&-7\\
0&1&-5&-4
\end{array}
\right|\\
\overset{r_2\longleftrightarrow r_3}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}
\left|\begin{array}{ccccccc}
1&1&1&3\\
0&1&-8&-7\\
0&-4&-7&-11\\
0&1&-5&-4
\end{array}
\right|
\overset{r_3+4r_2\\r_4-r_2}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}
\left|\begin{array}{ccccccc}
1&1&1&3\\
0&1&-8&-7\\
0&0&-39&-39\\
0&0&3&3
\end{array}
\right|
={\color{blue}0}
\]
\[
{D_a}_4=\left|\begin{array}{ccccccc}
1&1&2&1\\
3&-1&-1&-4\\
2&3&-1&-6\\
1&2&3&-4
\end{array}
\right|
\overset{r_2-3r_1\\r_3-2r_1\\r_4-r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
1&1&2&1\\
0&-4&-7&-7\\
0&1&-5&-8\\
0&1&1&-5
\end{array}
\right|\\
\overset{r_2\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}
\left|\begin{array}{ccccccc}
1&1&2&1\\
0&1&1&-5\\
0&1&-5&-8\\
0&-4&-7&-7
\end{array}
\right|
\overset{r_3+4r_2\\r_4+3r_2}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}
\left|\begin{array}{ccccccc}
1&1&2&1\\
0&1&1&-5\\
0&0&-6&-3\\
0&0&-3&-27
\end{array}
\right|\\
\overset{r_3\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}+}
\left|\begin{array}{ccccccc}
1&1&2&1\\
0&1&1&-5\\
0&0&-3&-27\\
0&0&-6&-3
\end{array}
\right|
\overset{r_4-2r_3}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
1&1&2&1\\
0&1&1&-5\\
0&0&-3&-27\\
0&0&0&51
\end{array}
\right|
={\color{blue}-153}
\]
\[
\color{purple}{
\left\{
\begin{split}
x_1&\,&&&&&&&&&=&-&1\\
&&&x_2&&&&&&&=&-&1\\
&&&&&&x_3&&&&=&&0\\
&\,&&&&&&&&x_4&=&&1
\end{split}
\right. }
\]
\(\overline{\hspace{15cm}}\)
\[
\color{green}{
\left\{
\begin{split}
2x_1&-&&x_2&+&3&x_3&+&2&x_4&=4\\
3x_1&+&3&x_2&+&3&x_3&+&2&x_4&=6\\
3x_1&-&2&x_2&-&&x_3&+&2&x_4&=6\\
3x_1&-&&x_2&+&3&x_3&-&&x_4&=6
\end{split}
\right. }
\]
\[
D_b=\left|\begin{array}{ccccccc}
2&-1&3&2\\
3&3&3&2\\
3&-2&-1&2\\
3&-1&3&-1
\end{array}
\right|=
\,3\left|\begin{array}{ccccccc}
2&-1&3&2\\
1&1&1&\frac{2}{3}\\
3&-2&-1&2\\
3&-1&3&-1
\end{array}
\right|\\
\overset{r_1\longleftrightarrow r_2}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}3\left|\begin{array}{ccccccc}
1&1&1&\frac{2}{3}\\
2&-1&3&2\\
3&-2&-1&2\\
3&-1&3&-1
\end{array}
\right|
\overset{r_2-2r_1\\r_3-3r_1\\r_4-3r_1}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}3\left|\begin{array}{ccccccc}
1&1&1&\frac{2}{3}\\
0&-3&1&\frac{2}{3}\\
0&-5&-4&0\\
0&-4&0&-3
\end{array}
\right|\\
={\color{red}+}3\cdot4\left|\begin{array}{ccccccc}
1&1&1&\frac{2}{3}\\
0&-3&1&\frac{2}{3}\\
0&-5&-4&0\\
0&1&0&\frac{3}{4}
\end{array}
\right|
\overset{r_2\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}12\left|\begin{array}{ccccccc}
1&1&1&\frac{2}{3}\\
0&1&0&\frac{3}{4}\\
0&-5&-4&0\\
0&-3&1&\frac{2}{3}
\end{array}
\right|\\
\overset{r_3+5r_2\\r_4+3r_2}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}12\left|\begin{array}{ccccccc}
1&1&1&\frac{2}{3}\\
0&1&0&\frac{3}{4}\\
0&0&-4&\frac{15}{4}\\
0&0&1&\frac{35}{12}
\end{array}
\right|
\overset{r_3\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}+}12\left|\begin{array}{ccccccc}
1&1&1&\frac{2}{3}\\
0&1&0&\frac{3}{4}\\
0&0&1&\frac{35}{12}\\
0&0&-4&\frac{15}{4}
\end{array}
\right|\\
\overset{r_4+4r_3}{\overline{\overline{\hspace{3cm}}}}
{\color{red}+}12\left|\begin{array}{ccccccc}
1&1&1&\frac{2}{3}\\
0&1&0&\frac{3}{4}\\
0&0&1&\frac{35}{12}\\
0&0&0&\frac{185}{12}
\end{array}
\right|={\color{blue}185}
\]
\[
{D_b}_1=\left|\begin{array}{ccccccc}
4&-1&3&2\\
6&3&3&2\\
6&-2&-1&2\\
6&-1&3&-1
\end{array}
\right|=
\,4\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
6&3&3&2\\
6&-2&-1&2\\
6&-1&3&-1
\end{array}
\right|\\
\overset{r_2-6r_1\\r_3-6r_1\\r_4-6r_1}{\overline{\overline{\hspace{3cm}}}}
\,4\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
0&\frac{9}{2}&-\frac{3}{2}&-1\\
0&-\frac{1}{2}&-\frac{11}{2}&-1\\
0&\frac{1}{2}&-\frac{3}{2}&-4
\end{array}
\right|
\overset{r_2\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}4\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
0&\frac{1}{2}&-\frac{3}{2}&-4\\
0&-\frac{1}{2}&-\frac{11}{2}&-1\\
0&\frac{9}{2}&-\frac{3}{2}&-1
\end{array}
\right|\\
\overset{r_3+r_2\\r_4-9r_2}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}4\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
0&\frac{1}{2}&-\frac{3}{2}&-4\\
0&0&-7&-5\\
0&0&12&35
\end{array}
\right|
\overset{r_4+r_3}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}4\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
0&\frac{1}{2}&-\frac{3}{2}&-4\\
0&0&-7&-5\\
0&0&5&30
\end{array}
\right|\\
\overset{r_3+5r_2\\r_4+3r_2}{\overline{\overline{\hspace{3cm}}}}
{\color{red}-}4\cdot5\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
0&\frac{1}{2}&-\frac{3}{2}&-4\\
0&0&-7&-5\\
0&0&1&6
\end{array}
\right|
\overset{r_3\longleftrightarrow r_4}{\overline{\overline{\hspace{3cm}}}}
{\color{red}+}20\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
0&\frac{1}{2}&-\frac{3}{2}&-4\\
0&0&1&6\\
0&0&-7&-5
\end{array}
\right|\\
\overset{r_4+7r_3}{\overline{\overline{\hspace{3cm}}}}
\,20\left|\begin{array}{ccccccc}
1&-\frac{1}{4}&\frac{3}{4}&\frac{1}{2}\\
0&\frac{1}{2}&-\frac{3}{2}&-4\\
0&0&1&6\\
0&0&0&37
\end{array}
\right|={\color{blue}370}
\]
\[
{D_b}_2=\left|\begin{array}{ccccccc}
2&4&3&2\\
3&6&3&2\\
3&6&-1&2\\
3&6&3&-1
\end{array}
\right|
\overset{l_2-2r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
2&0&3&2\\
3&0&3&2\\
3&0&-1&2\\
3&0&3&-1
\end{array}
\right|
=\color{blue}{0}
\]
\[
{D_b}_3=\left|\begin{array}{ccccccc}
2&-1&4&2\\
3&3&6&2\\
3&-2&6&2\\
3&-1&6&-1
\end{array}
\right|
\overset{l_3-2r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
2&-1&0&2\\
3&3&0&2\\
3&-2&0&2\\
3&-1&0&-1
\end{array}
\right|
=\color{blue}{0}
\]
\[
{D_b}_4=\left|\begin{array}{ccccccc}
2&-1&3&4\\
3&3&3&6\\
3&-2&-1&6\\
3&-1&3&6
\end{array}
\right|
\overset{l_4-2r_1}{\overline{\overline{\hspace{3cm}}}}
\left|\begin{array}{ccccccc}
2&-1&3&0\\
3&3&3&0\\
3&-2&-1&0\\
3&-1&3&0
\end{array}
\right|
=\color{blue}{0}
\]
\[
\color{green}{
\left\{
\begin{split}
x_1&\,&&&&&&&&&=&2\\
&&&x_2&&&&&&&=&0\\
&&&&&&x_3&&&&=&0\\
&\,&&&&&&&&x_4&=&0
\end{split}
\right. }
\]

就服你 /< /good