Question

4．已知矩阵A=$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$，求A¹⁰．

Answer 1

分析 根据矩阵的乘法分别求得A²，A³，A⁴，…，即可求得A¹⁰．

解答 解：A²=$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$=$[\begin{array}{l}{2}&{2}\\{2}&{2}\end{array}]$，
A³=$[\begin{array}{l}{2}&{2}\\{2}&{2}\end{array}]$$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$=$[\begin{array}{l}{4}&{4}\\{4}&{4}\end{array}]$=$[\begin{array}{l}{{2}^{2}}&{{2}^{2}}\\{{2}^{2}}&{{2}^{2}}\end{array}]$，
A⁴=$[\begin{array}{l}{4}&{4}\\{4}&{4}\end{array}]$$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$=$[\begin{array}{l}{8}&{8}\\{8}&{8}\end{array}]$=$[\begin{array}{l}{{2}^{3}}&{{2}^{3}}\\{{2}^{3}}&{{2}^{3}}\end{array}]$，
…
∴A¹⁰=$[\begin{array}{l}{{2}^{9}}&{{2}^{9}}\\{{2}^{9}}&{{2}^{9}}\end{array}]$．

点评 本题考查矩阵的乘法的意义，考查矩阵乘法的运算法则，考查计算能力，属于基础题．