Question

4．已知二階矩陣M有特征值λ=-1及對(duì)應(yīng)的一個(gè)特征向量$[\begin{array}{l}{1}\\{-3}\end{array}]$，且矩陣M對(duì)應(yīng)的變換將點(diǎn)（2，-1）變換成（3，1）．
（1）求矩陣M；
（2）求矩陣M的逆矩陣．

Answer 1

分析 （1）設(shè)M=$[\begin{array}{cc}a&b\\ c&d\end{array}\right.]$，由題意得：$[\begin{array}{cc}a&b\\ c&d\end{array}\right.]$$[\begin{array}{l}{1}\\{-3}\end{array}]$=-$[\begin{array}{l}{1}\\{-3}\end{array}]$，$[\begin{array}{cc}a&b\\ c&d\end{array}\right.]$$[\begin{array}{c}2\\-1\end{array}\right.]$=$[\begin{array}{c}3\\ 1\end{array}\right.]$，解得矩陣M，
（2）先求矩陣M的行列式，進(jìn)而可求其逆矩陣，

解答 （1）設(shè)M=$[\begin{array}{cc}a&b\\ c&d\end{array}\right.]$，由題意得：$[\begin{array}{cc}a&b\\ c&d\end{array}\right.]$$[\begin{array}{l}{1}\\{-3}\end{array}]$=-$[\begin{array}{l}{1}\\{-3}\end{array}]$，
即$\left\{\begin{array}{l}a-3b=-1\\ c-3d=3\end{array}\right.$  ①；…（3分）
$[\begin{array}{cc}a&b\\ c&d\end{array}\right.]$$[\begin{array}{c}2\\-1\end{array}\right.]$=$[\begin{array}{c}3\\ 1\end{array}\right.]$，即$\left\{\begin{array}{l}2a-b=3\\ 2c-d=1\end{array}\right.$  ②；…（5分）
由①②，得M=$[\begin{array}{cc}2&1\\ 0&-1\end{array}\right.]$…（8分）
（2）矩陣的行列式為$\left|\begin{array}{cc}2&1\\ 0&-1\end{array}\right|$=-2-0=-2，
∴求矩陣M的逆矩陣M^-1=$[\begin{array}{cc}\frac{1}{2}&\frac{1}{2}\\ 0&-1\end{array}\right.]$…（10分）

點(diǎn)評(píng) 本題以矩陣為載體，考查矩陣的逆矩陣，考查矩陣M的特征值，關(guān)鍵是求其行列式，正確寫(xiě)出矩陣M的特征多項(xiàng)式．