矩阵解方程组_矩阵和方程组

矩阵解方程组

方程组:

$$\begin{gathered} 3x+2y=7 \\ -6x+6y=6 \end{gathered}$$

用矩阵表示:

[ 3 2 − 6 6 ] ∗ [ x y ] = [ 7 6 ]

$$\begin{bmatrix} 3 & 2 \\ -6 & 6 \\ \end{bmatrix}$$ * $$\begin{bmatrix} x \\ y \\ \end{bmatrix}$$ = $$\begin{bmatrix} 7 \\ 6 \end{bmatrix}$$ [3−6​26​]∗[xy​]=[76​]

用字母表示

A = [ 3 2 − 6 6 ] A =

$$\begin{bmatrix} 3 & 2 \\ -6 & 6 \\ \end{bmatrix}$$ A=[3−6​26​]

B = [ 7 6 ] B =

$$\begin{bmatrix} 7 \\ 6 \\ \end{bmatrix}$$ B=[76​]

[ x y ] = B ∗ A − 1

$$\begin{bmatrix} x \\ y \\ \end{bmatrix}$$ = B*A^{-1} [xy​]=B∗A−1

求解

A − 1 = 1 30 [ 6 − 2 6 3 ] A^{-1} = \frac{1}{30}

$$\begin{bmatrix} 6 & -2 \\ 6 & 3 \\ \end{bmatrix}$$ A−1=301​[66​−23​]

[ x y ] = B ∗ A − 1 = 1 30 [ 30 60 ] = [ 1 2 ]

$$\begin{bmatrix} x \\ y \\ \end{bmatrix}$$ = B*A^{-1} = \frac{1}{30} $$\begin{bmatrix} 30 \\ 60 \\ \end{bmatrix}$$ = $$\begin{bmatrix} 1 \\ 2 \\ \end{bmatrix}$$ [xy​]=B∗A−1=301​[3060​]=[12​]