Question

10．已知$A=（\begin{array}{l}{1}&{0}&{1}&{0}\\{2}&{1}&{2}&{0}\\{0}&{0}&{1}&{0}\\{1}&{1}&{1}&{1}\end{array}）$，試用矩陣初等行變換法求A的逆矩陣．

Answer 1

分析 由B=[A丨I]，對A與I進(jìn)行完全相同的若干初等行變換，把A化為單位矩陣，將單位矩陣化為A^-1．

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

點評 本題考查矩陣初等行變換法求矩陣的逆矩陣，考查計算能力，屬于中檔題．