Question

10．已知$A=（\begin{array}{l}{1}&{0}&{1}&{0}\\{2}&{1}&{2}&{0}\\{0}&{0}&{1}&{0}\\{1}&{1}&{1}&{1}\end{array}）$，试用矩阵初等行变换法求A的逆矩阵．

Answer 1

分析 由B=[A丨I]，对A与I进行完全相同的若干初等行变换，把A化为单位矩阵，将单位矩阵化为A^-1．

解答 解：B=[A丨I]=$[\begin{array}{l}{1}&{0}&{1}&{0}&{1}&{0}&{0}&{0}\\{2}&{1}&{2}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}&{1}&{0}\\{1}&{1}&{1}&{1}&{0}&{0}&{0}&{1}\end{array}]$→$[\begin{array}{l}{1}&{0}&{1}&{0}&{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{1}&{-1}&{0}&{0}&{1}\end{array}]$→$[\begin{array}{l}{1}&{0}&{1}&{0}&{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}&{1}&{-1}&{0}&{1}\end{array}]$
→$[\begin{array}{l}{1}&{0}&{0}&{0}&{1}&{0}&{-1}&{0}\\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}&{1}&{-1}&{0}&{1}\end{array}]$，
逆矩阵A^-1=$[\begin{array}{l}{1}&{0}&{-1}&{0}\\{-2}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{1}&{-1}&{0}&{1}\end{array}]$．

点评 本题考查矩阵初等行变换法求矩阵的逆矩阵，考查计算能力，属于中档题．