行列式解二元一次方程式

行列式

定義與求值

定義：${\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc$

例如：

 
   
     
       
         
           |
           
             
               
                 2
               
               
                 3
               
             
             
               
                 1
               
               
                 4
               
             
           
           |
         
       
       =
       2
       ⋅
       4
       −
       3
       ⋅
       1
       =
       5
     
   
   {\displaystyle {\begin{vmatrix}2&3\\1&4\end{vmatrix}}=2\cdot 4-3\cdot 1=5}
 

${\begin{vmatrix}2&3\\1&4\end{vmatrix}}=2\cdot 4-3\cdot 1=5$

用行列式解二元一次方程式

方程式組為：${\begin{cases}a_{1}x+b_{1}y=c_{1}\\a_{2}x+b_{2}y=c_{2}\end{cases}}$

其解如下：

 
   
     
       x
       =
       
         
           
             |
             
               
                 
                   
                     
                       c
                       
                         1
                       
                     
                   
                   
                     
                       b
                       
                         1
                       
                     
                   
                 
                 
                   
                     
                       c
                       
                         2
                       
                     
                   
                   
                     
                       b
                       
                         2
                       
                     
                   
                 
               
             
             |
           
           
             |
             
               
                 
                   
                     
                       a
                       
                         1
                       
                     
                   
                   
                     
                       b
                       
                         1
                       
                     
                   
                 
                 
                   
                     
                       a
                       
                         2
                       
                     
                   
                   
                     
                       b
                       
                         2
                       
                     
                   
                 
               
             
             |
           
         
       
       ,
       
       y
       =
       
         
           
             |
             
               
                 
                   
                     
                       a
                       
                         1
                       
                     
                   
                   
                     
                       c
                       
                         1
                       
                     
                   
                 
                 
                   
                     
                       a
                       
                         2
                       
                     
                   
                   
                     
                       c
                       
                         2
                       
                     
                   
                 
               
             
             |
           
           
             |
             
               
                 
                   
                     
                       a
                       
                         1
                       
                     
                   
                   
                     
                       b
                       
                         1
                       
                     
                   
                 
                 
                   
                     
                       a
                       
                         2
                       
                     
                   
                   
                     
                       b
                       
                         2
                       
                     
                   
                 
               
             
             |
           
         
       
     
   
   {\displaystyle x={\frac {\left|{\begin{matrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}},\qquad y={\frac {\left|{\begin{matrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}}}
 

$x={\frac {\left|{\begin{matrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}},\qquad y={\frac {\left|{\begin{matrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}}$

或是直接套公式：

$$

 
   
     
       x
       =
       
         
           
             
               c
               
                 1
               
             
             
               b
               
                 2
               
             
             −
             
               c
               
                 2
               
             
             
               b
               
                 1
               
             
           
           
             
               a
               
                 1
               
             
             
               b
               
                 2
               
             
             −
             
               a
               
                 2
               
             
             
               b
               
                 1
               
             
           
         
       
     
   
   {\displaystyle x={\frac {c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}}}
 

$x={\frac {c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}}$,

$$

 
   
     
       y
       =
       
         
           
             
               a
               
                 1
               
             
             
               c
               
                 2
               
             
             −
             
               a
               
                 2
               
             
             
               c
               
                 1
               
             
           
           
             
               a
               
                 1
               
             
             
               b
               
                 2
               
             
             −
             
               a
               
                 2
               
             
             
               b
               
                 1
               
             
           
         
       
     
   
   {\displaystyle y={\frac {a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}}}
 

$y={\frac {a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}}$

例：

$$

 
   
     
       
         
           {
           
             
               
                 2
                 x
                 +
                 37
                 y
                 =
                 2
               
             
             
               
                 2
                 x
                 +
                 119
                 y
                 =
                 2
               
             
           
           
         
       
     
   
   {\displaystyle {\begin{cases}2x+37y=2\\2x+119y=2\end{cases}}}
 

${\begin{cases}2x+37y=2\\2x+119y=2\end{cases}}$

解：$$

 
   
     
       x
       =
       
         
           
             2
             ×
             119
             −
             2
             ×
             37
           
           
             2
             ×
             119
             −
             2
             ×
             37
           
         
       
       =
       
         
           
             238
             −
             74
           
           
             238
             −
             74
           
         
       
     
   
   {\displaystyle x={\frac {2\times 119-2\times 37}{2\times 119-2\times 37}}={\frac {238-74}{238-74}}}
 

$x={\frac {2\times 119-2\times 37}{2\times 119-2\times 37}}={\frac {238-74}{238-74}}$,

238－74=164, x = 1

$$

 
   
     
       y
       =
       
         
           
             4
             −
             4
           
           
             238
             −
             74
           
         
       
     
   
   {\displaystyle y={\frac {4-4}{238-74}}}
 

$y={\frac {4-4}{238-74}}$

4－4=0,238－74=164, y = 0