题目内容

2.已知矩阵A=$[\begin{array}{l}{-1}&{0}\\{0}&{2}\end{array}]$,B=$[\begin{array}{l}{1}&{2}\\{0}&{6}\end{array}]$,求矩阵A-1B.

分析 设矩阵A-1=$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$,通过AA-1为单位矩阵可得A-1,进而可得结论.

解答 解:设矩阵A的逆矩阵为$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$,
则$[\begin{array}{l}{-1}&{0}\\{0}&{2}\end{array}]$$[\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}]$=$[\begin{array}{l}{1}&{0}\\{0}&{1}\end{array}]$,即$[\begin{array}{l}{-a}&{-b}\\{2c}&{2d}\end{array}]$=$[\begin{array}{l}{1}&{0}\\{0}&{1}\end{array}]$,
故a=-1,b=0,c=0,d=$\frac{1}{2}$,
从而A-1=$[\begin{array}{l}{-1}&{0}\\{0}&{\frac{1}{2}}\end{array}]$,
∴A-1B=$[\begin{array}{l}{-1}&{0}\\{0}&{\frac{1}{2}}\end{array}]$$[\begin{array}{l}{1}&{2}\\{0}&{6}\end{array}]$=$[\begin{array}{l}{-1}&{-2}\\{0}&{3}\end{array}]$.

点评 本题考查逆矩阵、矩阵的乘法,考查运算求解能力,属于基础题.

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