例题 4.1

Question

1. 已知 $\mathbf{A}=\left(\begin{array}{ll}2& 0\\ 0& 3\end{array}\right)$, 则 ${\mathbf{A}}^{-1}=\left(\begin{array}{ll}\frac{1}{2}& 0\\ 0& \frac{1}{3}\end{array}\right)$.
2. 设 $\mathbf{A}=\left(\begin{array}{ll}2& 1\\ 0& 3\end{array}\right)$ ，求其逆矩阵。

解 (1) 由定义验算即得. (2) 设 $\mathbf{B}=\left(\begin{array}{ll}{b}_1& b_2\\ b_3& b_4\end{array}\right)$ 是 $\mathbf{A}$ 的逆矩阵, 则

$$\left(\begin{array}{ll}2& 1\\ 0& 3\end{array}\right)\left(\begin{array}{ll}{b}_1& b_2\\ b_3& b_4\end{array}\right)=\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right)$$

那么

$$\{\begin{array}{rl}2b_1+b_3& =1\\ 2b_2+b_4& =0\\ 3b_3& =0\\ 3b_4& =1\end{array}$$

解得 $b_1=\frac{1}{2},b_2=-\frac{1}{6},b_3=0,b_4=\frac{1}{3}$, 即 ${\mathbf{A}}^{-1}=\left(\begin{array}{rr}\frac{1}{2}& -\frac{1}{6}\\ 0& \frac{1}{3}\end{array}\right)$.