Question

3．已知点A（1，1）在矩阵$M=[{\begin{array}{l}1&a\\ 0&b\end{array}}]$对应的变换作用下得到点B（1，2），点B在矩阵$N=[{\begin{array}{l}m&{-1}\\ n&0\end{array}}]$对应的变换作用下得到点C（-2，1），求矩阵MN的逆矩阵．

Answer 1

分析 根据条件，先求出$M=[{\begin{array}{l}1&0\\ 0&2\end{array}}]$，再求出$N=[{\begin{array}{l}0&{-1}\\ 1&0\end{array}}]$，因此得出$MN=[{\begin{array}{l}1&0\\ 0&2\end{array}}][{\begin{array}{l}0&{-1}\\ 1&0\end{array}}]=[{\begin{array}{l}0&{-1}\\ 2&0\end{array}}]$，最后根据逆矩阵定义的得出${[{\begin{array}{l}0&{-1}\\ 2&0\end{array}}]^{-1}}=[{\begin{array}{l}0&{\frac{1}{2}}\\{-1}&0\end{array}}]$．

解答 解：由题意可得，$[{\begin{array}{l}1&a\\ 0&b\end{array}}][\begin{array}{l}1\\ 1\end{array}]=[\begin{array}{l}1+a\\ b\end{array}]=[\begin{array}{l}1\\ 2\end{array}]$，
解得a=0，b=2，所以$M=[{\begin{array}{l}1&0\\ 0&2\end{array}}]$，
又$[{\begin{array}{l}m&{-1}\\ n&0\end{array}}][\begin{array}{l}1\\ 2\end{array}]=[{\begin{array}{l}{m-2}\\ n\end{array}}]=[\begin{array}{l}-2\\ 1\end{array}]$，
解得m=0，n=1，所以$N=[{\begin{array}{l}0&{-1}\\ 1&0\end{array}}]$，
则$MN=[{\begin{array}{l}1&0\\ 0&2\end{array}}][{\begin{array}{l}0&{-1}\\ 1&0\end{array}}]=[{\begin{array}{l}0&{-1}\\ 2&0\end{array}}]$，
所以，矩阵MN的逆矩阵为${[{\begin{array}{l}0&{-1}\\ 2&0\end{array}}]^{-1}}=[{\begin{array}{l}0&{\frac{1}{2}}\\{-1}&0\end{array}}]$．

点评 本题主要考查了矩阵的运算以及逆矩阵的求解，涉及矩阵的运算法则和逆矩阵的定义，属于中档题．