행렬(Matrix) 9. 역행렬을 이용한 연립방정식 풀이

$n$ 개의 미지주 $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$에 관한 $m$ 개의 연립방정식

        $a_{11} x_{1} + a_{12} x_{2} + \cdots + a_{1n} x_{n} = b_{1} $

        $a_{21} x_{1} + a_{22} x_{2} + \cdots + a_{2n} x_{n} = b_{2} $

        $a_{m1} x_{1} + a_{m2} x_{2} + \cdots + a_{mn} x_{n} = b_{m} $

 

        $AX = B$

        $\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots &&& \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn}\end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n}\end{bmatrix} = \begin{bmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{m}\end{bmatrix}$

 

        $AX = B$

        $A^{-1} (AX) = A^{-1} B$       $(A^{-1} A = I , IX = X)$

        $X = A^{-1} B$

 

ex 1)

        $2x_{1} - 9 x_{2} = 15$

        $3x_{1} + 6 x_{2} = 16$

 

        $\begin{bmatrix} 2 & -9 \\ 3 & 6 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix} 15 \\ 16 \end{bmatrix}$        $\begin{vmatrix} 2 & -9 \\ 3 & 6 \end{vmatrix} = 39 \neq 0$ 정치행렬

        $\begin{bmatrix} 2 & -9 \\ 3 & 6 \end{bmatrix}^{-1} = \frac{1}{39} \begin{bmatrix} 6 & -3 \\ 9 & 2 \end{bmatrix}^{T} = \frac{1}{39} \begin{bmatrix} 6 & 9 \\ -3 & 2 \end{bmatrix}$

        $\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \frac{1}{39} \begin{bmatrix} 6 & 9 \\ -3 & 2 \end{bmatrix} \begin{bmatrix} 15 \\ 16 \end{bmatrix} = \frac{1}{39} \begin{bmatrix} 6 \times 15 + 9 \times 16 \\ -3 \times 15 + 2 \times 16 \end{bmatrix} = \begin{bmatrix} 6 \\ -1/3 \end{bmatrix}$

        ∴ $x_{1} = 6$,    $x_{2} = -1/3$

 

ex 2)

        $2x_{1} + x_{3} = 2$

        $5x_{1} + 5x_{2} + 6x_{3} = -1$

        $-2x_{1} + 3x_{2} + 4x_{3} = 4$

        $\Rightarrow \begin{bmatrix} 2 & 0 & 1 \\ 5 & 5 & 6 \\ -2 & 3 & 4 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \\ 4\end{bmatrix}$

 

        $A = \begin{bmatrix} 2 & 0 & 1 \\ 5 & 5 & 6 \\ -2 & 3 & 4 \end{bmatrix}$ ,    $A^{-1} = \begin{bmatrix} -2 & 5 & -3 \\ -8 & 17 & -10 \\ 5 & -10 & 6 \end{bmatrix}$

 

        $X=A^{-1} B$

        $\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} = \begin{bmatrix} -2 & 5 & -3 \\ -8 & 17 & -10 \\ 5 & -10 & 6 \end{bmatrix} \begin{bmatrix} 2 \\ 4 \\ -1 \end{bmatrix}$

                $\begin{bmatrix} (-2)(2) + (5)(4) + (-3)(-1) \\ (-8)(2) + (17)(4) + (-10)(-1) \\ (5)(2) + (-10)(4) + (6)(-1)\end{bmatrix} = \begin{bmatrix} 19 \\ 62 \\ -36\end{bmatrix}$    $\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}$