Question

18．请用逆矩阵的方法求二元一次方程组$\left\{\begin{array}{l}2x-y=1\\ 2y-x=1\end{array}\right.$的解．

Answer 1

分析 记A=$[\begin{array}{l}{2}&{-1}\\{-1}&{2}\end{array}]$，写出其逆矩阵，再由$[\begin{array}{l}{\frac{2}{3}}&{\frac{1}{3}}\\{\frac{1}{3}}&{\frac{2}{3}}\end{array}]$$[\begin{array}{l}{1}\\{1}\end{array}]$=$[\begin{array}{l}{1}\\{1}\end{array}]$，即可解得原方程组的解．

解答 解：记A=$[\begin{array}{l}{2}&{-1}\\{-1}&{2}\end{array}]$，则A^-1=$[\begin{array}{l}{\frac{2}{3}}&{\frac{1}{3}}\\{\frac{1}{3}}&{\frac{2}{3}}\end{array}]$．
两边左乘A^-1可得：X=A^-1•B=$[\begin{array}{l}{\frac{2}{3}}&{\frac{1}{3}}\\{\frac{1}{3}}&{\frac{2}{3}}\end{array}]$$[\begin{array}{l}{1}\\{1}\end{array}]$=$[\begin{array}{l}{1}\\{1}\end{array}]$，
所以，原方程组的解为$\left\{\begin{array}{l}{x=1}\\{y=1}\end{array}\right.$．

点评 本小题主要考查逆变换与逆矩阵的计算、系数矩阵的逆矩阵解方程组等基础知识，考查运算求解能力与转化思想．属于基础题．