Question

19．用逆矩阵的知识解方程MX=N，其中M=$|\begin{array}{l}{5}&{2}\\{4}&{1}\end{array}|$，N=$|\begin{array}{l}{5}\\{-8}\end{array}|$．

Answer 1

分析 先求出M^-1，利用X=M^-1N计算即可．

解答 解：∵M=$[\begin{array}{l}{5}&{2}\\{4}&{1}\end{array}]$，设M^-1=$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$，
∴MM^-1=$[\begin{array}{l}{1}&{0}\\{0}&{1}\end{array}]$，即$[\begin{array}{l}{5}&{2}\\{4}&{1}\end{array}]$$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$=$[\begin{array}{l}{1}&{0}\\{0}&{1}\end{array}]$，
∴$\left\{\begin{array}{l}{5a+2c=1}\\{5b+2d=0}\\{4a+c=0}\\{4b+d=1}\end{array}\right.$，解得M^-1=$[\begin{array}{l}{-\frac{1}{3}}&{\frac{2}{3}}\\{\frac{4}{3}}&{-\frac{5}{3}}\end{array}]$，
又∵N=$[\begin{array}{l}{5}\\{-8}\end{array}]$，MX=N，
∴X=M^-1N=$[\begin{array}{l}{-\frac{1}{3}}&{\frac{2}{3}}\\{\frac{4}{3}}&{-\frac{5}{3}}\end{array}]$$[\begin{array}{l}{5}\\{-8}\end{array}]$=$[\begin{array}{l}{-7}\\{20}\end{array}]$．

点评 本题考查矩阵相关知识，注意解题方法的积累，属于中档题．