Question

16．在平面直角坐标系xOy中，设点P（1，1）在矩阵$M=[{\begin{array}{l}1&a\\ b&4\end{array}}]$对应的变换下得到点Q（3，7），求M^-1．

Answer 1

分析 由矩阵的变换求得a和b的值，求得丨M丨及M*，即可求得M^-1．

解答 解：由$[\begin{array}{l}{1}&{a}\\{b}&{4}\end{array}]$$[\begin{array}{l}{1}\\{1}\end{array}]$=$[\begin{array}{l}{3}\\{7}\end{array}]$，
∴$\left\{\begin{array}{l}{1+a=3}\\{b+4=7}\end{array}\right.$，解得：$\left\{\begin{array}{l}{a=2}\\{b=3}\end{array}\right.$，
M=$[\begin{array}{l}{1}&{2}\\{3}&{4}\end{array}]$，
丨M丨=1×4-2×3=-2
M^-1=$\frac{1}{丨M丨}$×$[\begin{array}{l}{4}&{-2}\\{-3}&{1}\end{array}]$=$[\begin{array}{l}{-2}&{1}\\{\frac{3}{2}}&{-\frac{1}{2}}\end{array}]$，
M^-1=$[\begin{array}{l}{-2}&{1}\\{\frac{3}{2}}&{-\frac{1}{2}}\end{array}]$．

点评 本题考查矩阵的变换，考查求二阶矩阵的逆矩阵，考查计算能力，属于中档题．