Question

18．請(qǐng)用逆矩陣的方法求二元一次方程組$\left\{\begin{array}{l}2x-y=1\\ 2y-x=1\end{array}\right.$的解．

Answer 1

分析 記A=$[\begin{array}{l}{2}&{-1}\\{-1}&{2}\end{array}]$，寫(xiě)出其逆矩陣，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}]$，即可解得原方程組的解．

解答 解：記A=$[\begin{array}{l}{2}&{-1}\\{-1}&{2}\end{array}]$，則A^-1=$[\begin{array}{l}{\frac{2}{3}}&{\frac{1}{3}}\\{\frac{1}{3}}&{\frac{2}{3}}\end{array}]$．
兩邊左乘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}]$，
所以，原方程組的解為$\left\{\begin{array}{l}{x=1}\\{y=1}\end{array}\right.$．

點(diǎn)評(píng) 本小題主要考查逆變換與逆矩陣的計(jì)算、系數(shù)矩陣的逆矩陣解方程組等基礎(chǔ)知識(shí)，考查運(yùn)算求解能力與轉(zhuǎn)化思想．屬于基礎(chǔ)題．