Question

17．已知矩陣$A=[{\begin{array}{l}a&b\\ c&d\end{array}}]$，若矩陣A屬于特征值6的一個特征向量為$\overrightarrow{{α}_{1}}$=$[\begin{array}{l}{1}\\{1}\end{array}]$，屬于特征值1的一個特征向量為$\overrightarrow{{α}_{2}}$=$[\begin{array}{l}{3}\\{-2}\end{array}]$．求A的逆矩陣．

Answer 1

分析 根據(jù)矩陣特征值和特征向量的性質(zhì)代入列方程組，求得a、b、c和d的值，求得矩陣A，丨A丨及A*，由A^-1=$\frac{1}{丨A丨}$×A*，即可求得A^-1．

解答 解：矩陣A屬于特征值6的一個特征向量為$\overrightarrow{{α}_{1}}$=$[\begin{array}{l}{1}\\{1}\end{array}]$，
∴$[\begin{array}{l}{a}&\\{c}&nz1d7pt\end{array}]$$[\begin{array}{l}{1}\\{1}\end{array}]$=6$[\begin{array}{l}{1}\\{1}\end{array}]$，即$[\begin{array}{l}{a+b}\\{c+d}\end{array}]$=$[\begin{array}{l}{6}\\{6}\end{array}]$，
屬于特征值1的一個特征向量為$\overrightarrow{{α}_{2}}$=$[\begin{array}{l}{3}\\{-2}\end{array}]$．
∴$[\begin{array}{l}{a}&\\{c}&zprtd73\end{array}]$$[\begin{array}{l}{3}\\{-2}\end{array}]$=$[\begin{array}{l}{3}\\{-2}\end{array}]$，$[\begin{array}{l}{3a-2b}\\{3c-2d}\end{array}]$=$[\begin{array}{l}{3}\\{-2}\end{array}]$，
∴$\left\{\begin{array}{l}{a+b=6}\\{c+d=6}\\{3a-2b=3}\\{3c-2d=-2}\end{array}\right.$，解得：$\left\{\begin{array}{l}{a=3}\\{b=3}\\{c=2}\\{d=4}\end{array}\right.$，
矩陣A=$[\begin{array}{l}{3}&{3}\\{2}&{4}\end{array}]$，
丨A丨=$|\begin{array}{l}{3}&{3}\\{2}&{4}\end{array}|$=6，A*=$[\begin{array}{l}{4}&{-3}\\{-2}&{3}\end{array}]$，
A^-1=$\frac{1}{丨A丨}$×A*=$[\begin{array}{l}{\frac{2}{3}}&{-\frac{1}{2}}\\{-\frac{1}{3}}&{\frac{1}{2}}\end{array}]$，
∴A^-1=$[\begin{array}{l}{\frac{2}{3}}&{-\frac{1}{2}}\\{-\frac{1}{3}}&{\frac{1}{2}}\end{array}]$．

點評 本題考查矩陣的特征值及特征向量的性質(zhì)，考查逆矩陣的求法，考查計算能力，屬于中檔題．