Question

17．已知矩阵M对应的变换将点（-5，-7）变换为（2，1），其逆矩阵M^-1有特征值-1，对应的一个特征向量为$[{\begin{array}{l}1\\ 1\end{array}}]$，求矩阵M．

Answer 1

分析 根据矩阵的变换求得M$[\begin{array}{l}{-5}\\{-7}\end{array}]$=$[\begin{array}{l}{2}\\{1}\end{array}]$，利用矩阵的特征向量及特征值的关系，利用矩阵的乘法，即可求得M的逆矩阵，即可求得矩阵M．

解答 解：由题意可知：M$[\begin{array}{l}{-5}\\{-7}\end{array}]$=$[\begin{array}{l}{2}\\{1}\end{array}]$，
M^-1$[\begin{array}{l}{1}\\{1}\end{array}]$=$[\begin{array}{l}{-1}\\{-1}\end{array}]$，
∴M^-1$[\begin{array}{l}{2}\\{1}\end{array}]$=$[\begin{array}{l}{-5}\\{-7}\end{array}]$，
设M^-1=$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$，则$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$$[\begin{array}{l}{2}\\{1}\end{array}]$=$[\begin{array}{l}{-5}\\{-7}\end{array}]$，$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$$[\begin{array}{l}{1}\\{1}\end{array}]$=$[\begin{array}{l}{-1}\\{-1}\end{array}]$，
则$\left\{\begin{array}{l}{2a+b=-5}\\{2c+d=-7}\\{a+b=-1}\\{c+d=-1}\end{array}\right.$，解得：$\left\{\begin{array}{l}{a=-4}\\{b=3}\\{c=-6}\\{d=5}\end{array}\right.$，则M^-1=$[\begin{array}{l}{-4}&{3}\\{-6}&{5}\end{array}]$，
det（M^-1）=-20+18=-2，
则M=$[\begin{array}{l}{-\frac{5}{2}}&{\frac{3}{2}}\\{-3}&{2}\end{array}]$．
∴矩阵M=$[\begin{array}{l}{-\frac{5}{2}}&{\frac{3}{2}}\\{-3}&{2}\end{array}]$．

点评 本题考查矩阵及逆矩阵的求法，矩阵的乘法，矩阵的特征值及特征向量的关系，考查转化思想，属于中档题．