矩阵解方程组

矩阵解方程组

方程组:

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

用矩阵表示:

$$\left[\begin{array}{cc}3& 2\\ -6& 6\end{array}\right]\ast \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}7\\ 6\end{array}\right]$$

用字母表示

$$A=\left[\begin{array}{cc}3& 2\\ -6& 6\end{array}\right]$$

$$B=\left[\begin{array}{c}7\\ 6\end{array}\right]$$

$$\left[\begin{array}{c}x\\ y\end{array}\right]=B\ast A^{-1}$$

求解

$$A^{-1}=\frac{1}{30}\left[\begin{array}{cc}6& -2\\ 6& 3\end{array}\right]$$

$$\left[\begin{array}{c}x\\ y\end{array}\right]=B\ast A^{-1}=\frac{1}{30}\left[\begin{array}{c}30\\ 60\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]$$