Question

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

Answer 1

分析 由矩陣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的逆矩陣．

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

點(diǎn)評 本題考查逆矩陣的求法，是基礎(chǔ)題，解題時(shí)要認(rèn)真審題，注意行列式、伴隨矩陣、逆矩陣的性質(zhì)的合理運(yùn)用．