Question

20．利用逆矩陣解方程組$\left\{\begin{array}{l}{2x+y=8}\\{4x-5y=2}\end{array}\right.$．

Answer 1

分析 先根據(jù)系數(shù)行列式，得到矩陣A可逆．寫出其逆矩陣，即可解得原方程組的解

解答 解：先計算矩陣行列式的值$|\begin{array}{l}{2}&{1}\\{4}&{-5}\end{array}|$=-14
∴A^-1=$[\begin{array}{l}{\frac{5}{14}}&{\frac{1}{14}}\\{\frac{2}{7}}&{-\frac{1}{7}}\end{array}]$
方程組的解為$[\begin{array}{l}{x}\\{y}\end{array}]$=$[\begin{array}{l}{\frac{5}{14}}&{\frac{1}{14}}\\{\frac{2}{7}}&{-\frac{1}{7}}\end{array}]$$[\begin{array}{l}{8}\\{2}\end{array}]$=$[\begin{array}{l}{3}\\{2}\end{array}]$．

點評 本小題主要考查逆變換與逆矩陣的計算、系數(shù)矩陣的逆矩陣解方程組等基礎知識，考查運算求解能力與轉化思想．屬于基礎題．