Question

1．已知矩阵A=$[\begin{array}{l}{-2}&{1}\\{\frac{3}{2}}&{-\frac{1}{2}}\end{array}]$，则A的逆矩阵是$[\begin{array}{l}{1}&{2}\\{3}&{4}\end{array}]$．

Answer 1

分析 由矩阵A，求出|A|=-$\frac{1}{2}$，A^*=$[\begin{array}{l}{-\frac{1}{2}}&{-1}\\{-\frac{3}{2}}&{-2}\end{array}]$，再由A^-1=$\frac{1}{|A|}{A}^{*}$，能求出A的逆矩阵．

解答 解：∵矩阵A=$[\begin{array}{l}{-2}&{1}\\{\frac{3}{2}}&{-\frac{1}{2}}\end{array}]$，∴|A|=$|\begin{array}{l}{-2}&{1}\\{\frac{3}{2}}&{-\frac{1}{2}}\end{array}|$=-$\frac{1}{2}$，
∵A^*=$[\begin{array}{l}{-\frac{1}{2}}&{-1}\\{-\frac{3}{2}}&{-2}\end{array}]$，
∴A的逆矩阵A^-1=$\frac{1}{|A|}{A}^{*}$=$[\begin{array}{l}{1}&{2}\\{3}&{4}\end{array}]$．
故答案为：$[\begin{array}{l}{1}&{2}\\{3}&{4}\end{array}]$．

点评 本题考查逆矩阵的求法，是基础题，解题时要认真审题，注意行列式、伴随矩阵、逆矩阵的性质的合理运用．