Question

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

Answer 1

分析 先利用矩阵的乘法公式求出MN，由此能利用矩阵的初等变换能求出矩阵MN的逆矩阵．

解答 解：∵矩阵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}]$，
∴矩阵MN的逆矩阵是$[\begin{array}{l}{\frac{1}{3}}&{0}\\{0}&{2}\end{array}]$．
故答案为：$[\begin{array}{l}{\frac{1}{3}}&{0}\\{0}&{2}\end{array}]$．

点评 本题考查两个矩阵乘积的逆矩阵的求法，是中档题，解题时要认真审题，注意矩阵的乘法公式和矩阵的初等变换的合理运用．