Question

12．已知矩陣M=$[\begin{array}{l}{3}&{0}\\{0}&{1}\end{array}]$，N=$[\begin{array}{l}{1}&{0}\\{0}&{\frac{1}{2}}\end{array}]$，則矩陣MN的逆矩陣是$[\begin{array}{l}{\frac{1}{3}}&{0}\\{0}&{2}\end{array}]$．

Answer 1

分析 先利用矩陣的乘法公式求出MN，由此能利用矩陣的初等變換能求出矩陣MN的逆矩陣．

解答 解：∵矩陣M=$[\begin{array}{l}{3}&{0}\\{0}&{1}\end{array}]$，N=$[\begin{array}{l}{1}&{0}\\{0}&{\frac{1}{2}}\end{array}]$，
∴MN=$[\begin{array}{l}{3}&{0}\\{0}&{1}\end{array}]$$[\begin{array}{l}{1}&{0}\\{0}&{\frac{1}{2}}\end{array}]$=$[\begin{array}{l}{3}&{0}\\{0}&{\frac{1}{2}}\end{array}]$，
∵$[\begin{array}{l}{3}&{0}&{\;}&{1}&{0}\\{0}&{\frac{1}{2}}&{\;}&{0}&{1}\end{array}]$→$[\begin{array}{l}{1}&{0}&{\;}&{\frac{1}{3}}&{0}\\{0}&{\frac{1}{2}}&{\;}&{0}&{1}\end{array}]$→$[\begin{array}{l}{1}&{0}&{\;}&{\frac{1}{3}}&{0}\\{0}&{1}&{\;}&{0}&{2}\end{array}]$，
∴矩陣MN的逆矩陣是$[\begin{array}{l}{\frac{1}{3}}&{0}\\{0}&{2}\end{array}]$．
故答案為：$[\begin{array}{l}{\frac{1}{3}}&{0}\\{0}&{2}\end{array}]$．

點評 本題考查兩個矩陣乘積的逆矩陣的求法，是中檔題，解題時要認(rèn)真審題，注意矩陣的乘法公式和矩陣的初等變換的合理運用．