Question

10．已知矩阵A=$[\begin{array}{l}{1}&{1}\\{1}&{2}\end{array}]$，且AB=$[\begin{array}{l}{1}&{0}\\{0}&{1}\end{array}]$，则矩阵B=$[\begin{array}{l}{2}&{-1}\\{-1}&{1}\end{array}]$．

Answer 1

分析 利用初等变换法计算即得结论．

解答 解：∵A=$[\begin{array}{l}{1}&{1}\\{1}&{2}\end{array}]$，
∴$[\begin{array}{l}{1}&{1}&{1}&{0}\\{1}&{2}&{0}&{1}\end{array}]$→$[\begin{array}{l}{1}&{1}&{1}&{0}\\{0}&{1}&{-1}&{1}\end{array}]$→$[\begin{array}{l}{1}&{0}&{2}&{-1}\\{0}&{1}&{-1}&{1}\end{array}]$，
∴A^-1=$[\begin{array}{l}{2}&{-1}\\{-1}&{1}\end{array}]$，
∵AB=$[\begin{array}{l}{1}&{0}\\{0}&{1}\end{array}]$，
∴B=A^-1=$[\begin{array}{l}{2}&{-1}\\{-1}&{1}\end{array}]$，
故答案为：$[\begin{array}{l}{2}&{-1}\\{-1}&{1}\end{array}]$．

点评 本题考查矩阵与逆矩阵，注意解题方法的积累，属于基础题．