Question

17．已知矩陣M對(duì)應(yīng)的變換將點(diǎn)（-5，-7）變換為（2，1），其逆矩陣M^-1有特征值-1，對(duì)應(yīng)的一個(gè)特征向量為$[{\begin{array}{l}1\\ 1\end{array}}]$，求矩陣M．

Answer 1

分析 根據(jù)矩陣的變換求得M$[\begin{array}{l}{-5}\\{-7}\end{array}]$=$[\begin{array}{l}{2}\\{1}\end{array}]$，利用矩陣的特征向量及特征值的關(guān)系，利用矩陣的乘法，即可求得M的逆矩陣，即可求得矩陣M．

解答 解：由題意可知：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$[\begin{array}{l}{2}\\{1}\end{array}]$=$[\begin{array}{l}{-5}\\{-7}\end{array}]$，
設(shè)M^-1=$[\begin{array}{l}{a}&\\{c}&dbtvxfx\end{array}]$，則$[\begin{array}{l}{a}&\\{c}&xatacp1\end{array}]$$[\begin{array}{l}{2}\\{1}\end{array}]$=$[\begin{array}{l}{-5}\\{-7}\end{array}]$，$[\begin{array}{l}{a}&\\{c}&mfw7nbd\end{array}]$$[\begin{array}{l}{1}\\{1}\end{array}]$=$[\begin{array}{l}{-1}\\{-1}\end{array}]$，
則$\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.$，則M^-1=$[\begin{array}{l}{-4}&{3}\\{-6}&{5}\end{array}]$，
det（M^-1）=-20+18=-2，
則M=$[\begin{array}{l}{-\frac{5}{2}}&{\frac{3}{2}}\\{-3}&{2}\end{array}]$．
∴矩陣M=$[\begin{array}{l}{-\frac{5}{2}}&{\frac{3}{2}}\\{-3}&{2}\end{array}]$．

點(diǎn)評(píng) 本題考查矩陣及逆矩陣的求法，矩陣的乘法，矩陣的特征值及特征向量的關(guān)系，考查轉(zhuǎn)化思想，屬于中檔題．